Invariant Manifolds, Passage through Resonance, Stability and a Computer Assisted Application to a Synchronous Motor

Diss. ETH No. 12744 June 1998

Invariant Manifolds, Passage through Resonance, Stability

and a

Computer Assisted Application to a Synchronous Motor

a Dissertation

submitted to the

Swiss Federal Institute of Technology Zurich

for the degree of

Doctor of Mathematics

presented by

Diego Giuseppe Tognola

Dipl. Math., University of Zurich

born July 3, 1967

citizen of Windisch AG, Switzerland

accepted on the recommendation of

Prof. Dr. U. Kirchgraber, referee

Prof. Dr. E. Zehnder, co–referee

PD Dr. D. Stoffer, co–referee

dedicated to all my friends

and

everyone supporting me during this work

Contents

Introduction v

1 Reduction to a Planar System 1

2 Averaging and Passage through Resonance in Plane Systems 69

3 The Stability of the Set h = 0 in Action Angle Coordinates 111

4 Application to a Miniature Synchronous Motor 125

5 Application to Van der Pol’s Equation 229

i

Kurzfassung

Ziel dieser Arbeit ist die Untersuchung eines Systems gewohnlicher Differentialgleichungen, welches einenMiniatur–Synchronmotor modelliert. Dieses System ist ein Spezialfall eines allgemeineren Problems,welches eigenstandig von mathematischem Interesse ist. Aus diesem Grunde wird der erste Teil dieserArbeit in einem abstrakter Rahmen durchgefuhrt. Der zweite Teil zeigt darauf die Anwendung auf dasphysikalische Problem.

Das erste Kapitel behandelt ein hamiltonsches und ein exponentiell stabiles lineares System, welchedurch schwache periodische Storungen gekoppelt sind. Das hamiltonsche System mit einem Freiheitsgradbesitze einen elliptischen Fixpunkt im Ursprung. Im ungestorten Fall besitze der Ursprung eine attraktiveZentrumsmannigfaltigkeit sowie eine stabile Mannigfaltigkeit. Unter Verwendung der Theorie invarianterMannigfaltigkeiten weisen wir nach, dass diese Struktur im wesentlichen bestehen bleibt. Die Diskussionauf die zeitabhangige, attraktive invariante Mannigfaltigkeit einschrankend, schliesst das erste Kapitelmit zwei verschiedenen Darstellungen des resultierenden reduzierten Systems.

Das zweite Kapitel zielt auf eine globale Diskussion des reduzierten Systems ab. Mittelungsmethodenwerden angewendet, um das Problem zu vereinfachen. Wir setzen voraus, dass nur endlich viele Resonan-zen existieren und teilen den Phasenraum in Nichtresonanz- und Resonanzzonen (sog. aussere und innereZonen). Die Nichtresonanzzonen bestehen aus dem ganzen Phasenraum, ausser kleinen Umgebungender Resonanzen. In den Resonanzzonen, welche kleine Umgebungen der Resonanzen abdecken, werdenKriterien fur strikte und fast–strikte Resonanzdurchgange hergeleitet. Fast–strikter Resonanzdurchgangbedeutet Durchgang aller Losungen, mit der moglichen Ausnahme einer asymptotisch kleinen Menge vonLosungen, welche in die Resonanz eingefangen werden konnen. Die angewendeten Mittelungsmethodenin der Nichtresonanzzone sind unublich und erlauben es, die zwei Zonen in einer Weise zu wahlen, sodasssie uberlappen.

Kapitel drei behandelt die zweite Darstellung des reduzierten Systems und ist fur die Stabilitatsdis-kussion passend. Mit Hilfe der Theorie von Floquet gewinnen wir eine Darstellung, welche es erlaubt,(In)Stabilitat umgehend zu diskutieren. Die Abhandlung deckt auch den degenerierten Fall, in welchemdie (In)Stabilitat nicht durch lineare Terme verursacht wird.

Der zweite Teil der Arbeit zeigt die Anwendung der im ersten Teil hergeleiteten Methoden und Resultate.Fur das Problem des Synchronmotors werden explizite Naherungen der relevanten Grossen analytischhergeleitet und numerisch ausgewertet. Die theoretischen Schlusse auf die Dynamik des Motors werdendurch numerische Simulationen bestatigt. Es wird gezeigt, dass sich der Motor dem Zustand der stabilen,synchronen Drehung, moduliert durch die zweite Harmonische, nahert, wenn er gestartet wird. WeitereEffekte wie der Einfluss mechanischer Reibung and zusatzlichem Drehmoment werden diskutiert. Strikterund fast–strikter Resonanzdurchlauf wird fur gewisse Parameter nachgewiesen.

ii

Abstract

The aim of this paper is to study a system of ordinary differential equations, modelling a miniaturesynchronous motor. This system is a special case of a more general problem which is of mathematicalinterest in itself. Hence an abstract framework is introduced in the first part of this work. The secondpart then presents the application to the physical problem.

Chapter one treats a Hamiltonian and an exponentially stable linear system, the two being coupled byweak periodic perturbations. The Hamiltonian system is of one degree of freedom and admits an ellipticfixed point at the origin. In the unperturbed case the origin admits an attractive center manifold as wellas a stable manifold. Using invariant manifold theory we establish that this structure essentially persists.Restricting the discussion to the time–dependant attractive invariant manifold, the first chapter closeswith two different representations of the resulting reduced system.

Chapter two aims at a global discussion of this reduced system. Averaging techniques are applied tosimplify the problem. We assume that there exist finitely many resonances only and split the phase spaceinto non–resonance and resonance zones (so–called outer and inner zones). The non–resonance zoneconsists of the entire phase space except small neighbourhoods of the resonances. In the resonance zone,which cover small neighbourhoods of the resonances, criteria for strict and almost strict passage throughresonances are deduced. Almost strict passage means passage of all solutions with the possible exceptionof an asymptotically small set of solutions which may be captured into the resonance. The averagingmethod applied in the non–resonance zone is non–standard and permits to choose the two regions in sucha way that they overlap.

Chapter three deals with the second representation of the reduced system and is suitable for the stabilitydiscussion. Using Floquet’s theory we gain a representation which permits to discuss (in)stability at once.The treatise covers the degenerate case where (in)stability is not caused by linear terms, as well.

The second part of the paper presents the application of the methods and results derived in part one.For the problem of the miniature synchronous motor, explicit approximations of the relevant quantitiesare deduced analytically and evaluated numerically. The theoretical conclusions on the dynamics ofthe motor are confirmed by numerical simulation. The motor is shown to approach the stable state ofsynchronous rotation with a small modulation by a second harmonic, when started. Additional effectssuch as the influence of mechanical friction and an additional torque are discussed. Strict and almoststrict passage through resonance is established for certain parameters.

iv

Introduction

The aim of this work is to study a particular type of miniature synchronous motor. Conventional syn-chronous motors are characterized by the property that under working conditions the rotor exhibits astable rotation, the frequency being that of the power supply (hence the term ”synchronous”). In or-der to enter the working conditions after switching on the motor, different techniques are suggested inelectrical engineering. Some of these techniques (such as pony motors, induction cages or electronic con-trols) are rather complicated. Hence in many papers the transient behaviour upon start and the stateof synchronous rotation are treated separately. By contrast, the type of motor considered here featuresa simple mechanism which permits a satisfactory physical modelling covering the entire process. Thismodel has been used by the manufacturer [12] for numerical studies and was presented in a colloquiumtalk in the nineteen 80’s. It is represented by the following non–linear time–periodic system of ordinarydifferential equations

d2

dτ2ϑ = −λ

J

√

i21 + i22 sin(ϕ)− ˜d

dτϑ− m

U0 sin(ωτ) = R i1 + Ld

dτi1 + λ

d

dτsin(ϑ)


dτi2 + λ

d

dτcos(ϑ) + u

d

dτu = i2/C.

(1)

The quantity ϑ is the angle of the rotor with respect to a fixed axis, i1 i2 correspond to the currents intwo parallel circuits and u describes the voltage of a condenser attached to the second power circuit.

Our approach for a mathematical treatise is based on perturbation theory. After some preliminarytransformations and assumptions on the parameters, the system turns out to be a special case of thefollowing problem

(q, p) = J∇H(q, p) + F (q, p, η, t, ε)

η = Aη +G(q, p, t, ε).(1.1)

This generalized problem consists of a one–degree of freedom Hamiltonian system (corresponding to themathematical pendulum in the above set of equations) and a linear system, the two being coupled bysmall periodic perturbations. As this system is of interest by itself, we introduce a general frameworkwhich might be of use elsewhere, too. The hypotheses we make reflect some of the features of the originalphysical problem, however. As to the Hamiltonian system these assumptions include the existence of anelliptic fixed point at the origin satisfying a non–resonance condition, as well as the existence of domainsfoliated by periodic solutions (such as the oscillatory and rotatory solutions of the pendulum equation).The matrix A is assumed to be exponentially asymptotically stable and the Fourier series with respectto time t of the 2 π–periodic perturbations F and G are assumed to be finite.

v

vi Introduction

The original physical problem suggests two main questions. The discussion of the state of synchronousrotation which is related to the existence and stability of a periodic solution near (q, p, η) = 0 and henceis local in nature. On the other hand, solutions describing the transition from start to stationary rotationare of upmost interest. They require a more global treatment. In a first part of this work a number ofkey results for systems of type (1.1) are derived which prove to be a tool kit for concrete applications. Ina second part these results are applied to the miniature synchronous motor.

The first part is split into three self–contained chapters. In chapter 1 it is shown that the fixed point(q, p, η) = 0 of the unperturbed system generates a unique 2 π–periodic solution. A discussion of itsstability is postponed until chapter 3. A time–dependant shift of the coordinates first yields a problemwhich is again of type (1.1), i.e.

( ˙Q, ˙P ) = J∇H(Q, P ) + F (Q, P ,H, t, ε)

H = AH+ G(Q, P , t, ε),(1.16)

but satisfies F (0, 0, 0, t, ε) = 0 and G(0, 0, t, ε) = 0. For ε = 0, the (Q, P )–plane H = 0 corresponds tothe center manifold of the origin, whereas the H–axis (Q, P ) = (0, 0) represents the stable manifold. Forε 6= 0 sufficiently small we establish the existence of an integral manifold (Q, P ) = V(t,H, ε), the so–calledstrongly stable manifold. This is achieved by adapting a result of Kelley [8] to our situation. Applyingthe transformation (Q, P ) = (Q,P ) + V(t,H, ε) then yields a system of the form

(Q, P ) = J∇H(Q,P ) + F (Q,P,H, t, ε)

H = AH+ G(Q,P,H, t, ε),(1.87)

where in particular F vanishes on the new H–axis, i.e.

F (0, 0,H, t, ε) = 0 G(0, 0, 0, t, ε) = 0. (1.88)

In a next step we replace (Q,P ) by action–angle coordinates (ϕ, h) ∈ R2. The transformed system isequivalent to (1.87) if we restrict (Q,P ) to regions of periodic solutions of (Q, P ) = J∇H(Q,P ). In viewof (1.88) such a region may be a neighbourhood of the fixed point (Q,P,H) = (0, 0, 0) as well. In thiscase, the set (h,H) = (0, 0) corresponds to (Q,P,H) = (0, 0, 0) and is invariant. The stability discussionof (h,H) = (0, 0) therefore yields information on the stability of (Q,P,H) = (0, 0, 0) which eventuallycorresponds to synchronous rotation in the case of our model of a synchronous motor. In action–anglecoordinates the system is of the form

ϕ = ω(h) + f(t, ϕ, h,H, ε)

h = g(t, ϕ, h,H, ε)

H = AH+ h(t, ϕ, h,H, ε)

(1.110)

where A still denotes the matrix introduced in (1.1) and f , g, h vanish for ε = 0. The unperturbedproblem corresponding to (1.110) suggests the existence of an attractive invariant manifold near H = 0.The majority of results on the existence of such manifolds (see e.g. Fenichel [4], Hirsch, Pugh, Shub[6]) are based on a discussion of Lyapunov type numbers of solutions. For the purpose of this workan approach based on more easily accessible quantities is more convenient, see Kirchgraber [9]. In thiswork we apply an adaption by Nipp/Stoffer [13] where the assumptions are expressed in terms of thevector field. It is here where the introduction of action–angle coordinates turns out to be advantageous.The attractive invariant manifold we establish admits the representation H = S(t, ϕ, h, ε) with S = 0for ε = 0. Since all solutions of (1.110) approach the invariant manifold, the discussion then reduces tothe reduced system, i.e. the restriction of eq. (1.110) to the attractive invariant manifold. This reducedsystem is two–dimensional but non-autonomous. It is represented in two different forms, either of whichwill be used in chapter 2 and chapter 3, respectively.

Introduction vii

The first representation of the reduced system given in chapter 1 is used for the global discussion. Takinginto account some additional properties of the original physical problem, chapter 2 deals with a systemof the form

ϕ = ω(h) +

3∑

j=2

εj∑

k,n∈Z

f jk,n(h) ei(kϕ+nt) + ε4 f4(t, ϕ, h, ε)

h =

3∑

j=2

εj∑

k,n∈Z

gjk,n(h) ei(kϕ+nt) + ε4 g4(t, ϕ, h, ε)

(2.1)

defined for ϕ, h ∈ R. Given km, nm ∈ Z and hm ∈ R such that gjkm,nm 6= 0 (j ∈ 2, 3) and km ω(hm) +nm = 0 the value hm is called a resonance. We assume that the set of resonances hm is finite. Moreover,for every resonance hm we require that d

dhω(hm) 6= 0 holds. In order to obtain information on thequalitative behaviour of (2.1) averaging techniques are applied. More precisely we apply time–dependantnear–identity transformations of the form h = h + O(ε2). This change of coordinates is defined in astandard way, see Kirchgraber [11] or Sanders/Verhulst [17]. We use it in a somewhat different way,however. As the transformation is singular in every resonance, it is applied outside a neighbourhood ofthe resonances. In order to keep the higher order terms small, the size of the neighbourhood of eachresonance must be chosen appropriately. We show that the neighbourhoods omitted may be chosen to beO(ε)–small. More precisely, for fixed δ > 0 and choosing |ε| < εO(δ) the transformation may be applied

outside |ε|δ –neighbourhoods of the resonances. In this outer region the transformed system then takes the

form

ϕ = ω(h) +O(ε)

˙h = ε2 g20,0(h) + ε2 δ2 g2(t, ϕ, h, ε, δ) +O(ε3).(2.23)

where g2 is still bounded. If on a subset of the outer region the map∣

∣g20,0∣

∣ is bounded from below, the

parameters δ and |ε| < εO(δ) may be chosen such that∣

∣

∣

˙h∣

∣

∣ > 0 and thus all solutions leave this subset.

Away from zeroes of g20,0 the qualitative behaviour is therefore determined simply by the sign of g20,0.

In the inner region, i.e. if h satisfies |h− hm| < 4 |ε|δ , a different near–identity change of coordinates is

defined. The resulting system then reads as follows

ϕ = ω(h) +O(ε2)

˙h = ε2 g20,0(h) + ε2∑

l∈N∗

g2lkm,lnm(h) eil(kmϕ+nmt) +O(ε3). (2.25)

Introducing the inner variables

ε h := const(

h− hm)

∀∣

∣h− hm∣

∣ < 4|ε|δ

ψ := km ϕ+ nm t,

(2.28)

and taking into account again some special features which arise in the application of the synchronousmotor, the system takes the form of a km 2 π–periodically perturbed pendulum with external torque,i.e. it is given by

ψ = ε h+ ε2 f2(t, ψ, h, ε)

˙h = ε (a0 + ac1 cos(ψ) + as1 sin(ψ)) + ε2 g2(t, ψ, h, ε).

(2.29)

The quantities a0, ac1 and as1 are determined by the Fourier coefficients g20,0 and g2km,nm evaluated at

h = hm.

viii Introduction

We then treat the following two situations:

1. |a0| >√

(ac1)2+ (as1)

2: For all solutions of the unperturbed system (i.e. (2.29) with the O(ε2)–

terms dropped) the quantity∣

∣

∣

˙h∣

∣

∣ is bounded from below. For ε sufficiently small, we conclude that

all solutions of (2.29) leave the region∣

∣h− hm∣

∣ < 4 |ε|δ . This behaviour is refered to as passage

through resonance.

2. |a0| <√

(ac1)2 + (as1)

2: The unperturbed system admits a hyperbolic and an elliptic fixed point on

the axis h = 0, generating periodic solutions for (2.29). It then is possible that solutions starting

near the boundary∣

∣h− hm∣

∣ = 4 |ε|δ are caught near h = 0 as t → ∞. This effect is called capture

into resonance. Here it is shown, however, that the set of such solutions has size O(ε).

By consequence, the global qualitative behaviour of most solutions is known, once the values of g20,0 andg2km,nm at h = hm are known. In chapter 4 the computation of these quantities will be the main point ofinterest.

In chapter 3 we consider a system of the form

ϕ = Ω0 + f ,0(t, ϕ, ε) + P(h) f ,1(t, ϕ, ε) + P(h)2f ,2(t, ϕ,P(h), ε)

h = P(h)ddhP(h)

g,1(t, ϕ, ε) + P(h)2

ddhP(h)

g,2(t, ϕ, ε) + P(h)3

ddhP(h)

g,3(t, ϕ,P(h), ε),(3.1)

according to the second representation of the reduced system introduced in chapter 1. The use of ananalytical cutting function P in (3.1) is reminiscent of the way in which action–angle coordinates wereintroduced. One may set P(h) = h in a neighbourhood of h = 0. We assume Ω0 6∈ 1

2Z and that f ,0(t, ϕ, ε),g,1(t, ϕ, ε) admit the following Fourier representation with respect to ϕ

f ,0(t, ϕ, ε) = f ,00 (t, ε) + f ,0c (t, ε) cos(2ϕ) + f ,0s (t, ε) sin(2ϕ)

g,1(t, ϕ, ε) = g,10 (t, ε)− f ,0s (t, ε) cos(2ϕ) + f ,0c (t, ε) sin(2ϕ).(3.2)

The maps f ,1, g,2 are assumed to be π–antiperiodic (i.e. f ,1(t, ϕ + π, ε) = −f ,1(t, ϕ, ε)). With the helpof Floquet’s theory we derive a near–identity transformation of the form

ϕ = ψ + u(t, ψ, ε) P(h) = rv(t, ε)

√

1 + ∂ψu(t, ψ, ε)(3.4)

transforming (3.1) to the form

ψ = Ω(ε) +O(r)

r = r g,10,0(ε) + r2 g,2(t, ψ, ε) + r3 g,3(t, ψ, r, ε)(3.20)

where Ω(0) = Ω0. Hence the coefficient g,10,0(ε) provides a criterion for r = 0 (and thus h = 0)to be asymptotically stable, or unstable, respectively. The quantity g,10,0(ε) will be evaluated in chapter4, in order to prove asymptotic stability of the periodic solution near (q, p, η) = (0, 0, 0) in case of thesynchronous motor problem.

Introduction ix

In the second part, chapter 4, we present the application of part one to the model of a miniaturesynchronous motor mentioned before. After some preliminary preparations eq. (1) is transformed into

q = p

p = −(a

2

)2

sin(q) + ε (η1 cos(q + t)− η2 sin(q + t))− ε2 p− ε2 (m+ )

η1 = −η1 + ε sin(q + t)

η2 = −η2 − 2 η3 + ε cos(q + t)

η3 = η2 − ε cos(q + t).

(4.14)

The quantity a is rougly equal to√

λR . For fixed a the perturbation parameter ε is given by a

√

λU0

.

Here we assume that the voltage U0 of the power supply and the moment of inertia J of the motor areproportional. Thus, ε tends to 0 provided U0 (and thus J) increases, while the magnetic dipol λ, and theresistance R are kept fixed. By consequence, the effect of induction generated by the rotating permanentmagnet and exerted on the coils decreases as ε→ 0.

In order to obtain preliminary insight into the features of (4.14) we present the results of various nu-merical simulations carried out with the help of the package dstool [3]. The results found confirm theanalytical discussion given later in this chapter. In addition, they demonstrate that the behaviour in aneighbourhood of the separatrix of the unperturbed problem of (4.14) is of no particular interest if a islarge. (Since the techniques introduced in part one rely on regions of periodic solutions of the Hamiltoniansystem, the neighbourhood of a separatrix is not covered by our analytical approach.)

The main task in chapter 4 is to apply the tools of part one to system (4.14) and to compute the keyquantities g20,0(h), g

2km,nm

(h) and g,10,0(ε). Among other things this amounts to explicitely constructsuitable approximations of the invariant manifolds introduced in chapter 1. The introduction of action–angle coordinates associated with the pendulum equation, is based on Fourier series of Jacobian ellipticfunctions. Eventually g20,0(h) and g2km,nm(h) are represented with the help of convolutions of Fourierseries. The complexity of this procedure requires the use of a software package for symbolic and numericalcomputations. The author has chosen the Maple [15] software package. Its synthax is simple and legiblefor readers with basic knowledge in programming. Hence the source code listed is comprehensible to agrowing community. For various choices of the parameters the dynamics of the model is discussed in termsof the physical behaviour of the motor. The influence of a mechanical friction (given by the parameter˜) and an external torque (m) is considered as well. Both situations considered in chapter 2, i.e. thecase of passage of all solutions up to an O(ε)–set as well as the passage of strictly all solutions throughresonances are established. The periodic solution near the origin, corresponding to the synchronousrotation of the shaft, is shown to be stable for all choices of the parameters. Moreover additional resultsare established: the possibility of asynchronous rotations, the modulation of the synchronous rotationstate by a second harmonic as well as a synchronous rotation with large variation of the angular speed(caused by a capture into resonance). The overall conclusion is that for sufficiently large a the motorbehaves favouritely, i.e. enters the state of stable synchronous rotation when switched on.

Chapter 4 closes with a result on the separatrix region for sufficiently small values of the parametera. In this situation, the existence of a global attractive invariant manifold of (4.14) is established. Thecorresponding reduced system then is of periodically perturbed pendulum type. Although an approximaterepresentation of the reduced system is not available, the construction of an approximate Melnikovfunction is feasible. The numerical evaluation of the corresponding formula then confirms the resultsfound by numerical simulation. More precisely, it is established that solutions starting with a frequencylarger than the frequency of the power supply may either enter the state of synchronous rotation or thefrequency may eventually tend to zero.

Chapter 1

Reduction to a Planar System

1.1 The System under Consideration

1.1.1 The Differential Equations

In this chapter we consider autonomous ordinary differential equations with a nonautonomous time–periodic perturbation. For the unperturbed case we assume two independent subsystems, a Hamiltoniansystem of one degree of freedom and a stable linear system. More precisely we will discuss equations ofthe form

(q, p) = J∇H(q, p) + F (q, p, η, t, ε)

η = Aη +G(q, p, t, ε),(1.1)

where (q, p) ∈ R2, η ∈ Rd and J :=[

0−1

10

]

represents the symplectic normal form.

We assume that A ∈ Rd×d has only eigenvalues on the left complex halfplane. The Hamiltonian H isassumed to be of class Cω (i.e. analytical), the maps F , G are assumed to be Cω, 2π–periodic withrespect to the time–variable t and vanishing as ε→ 0.

1

2 Chapter 1. Reduction to a Planar System

1.1.2 General Assumptions on the System

In this chapter we assume the following statements to be true

GA 1.1. The unperturbed Hamiltonian system

(q, p) = J∇H(q, p) (1.2)

satisfies the following set of assumptions :

(a) System (1.2) admits an elliptic fixed point in the origin. More precisely we assume thatin this situation ∇H(0, 0) = 0, ∂q∂pH(0, 0) = 0 and ∂2qH(0, 0), ∂2pH(0, 0) > 0. Moreover

D3H(0, 0) = 0 holds and Ω0 :=√

∂2qH(0, 0) ∂2pH(0, 0) 6∈ N := 0, 1, 2, . . ..

(b) There exist an interval J = (Jl,Jr) together with a mapping Ω ∈ Cω(J ,R) such that thesolution (q, p)(t; 0, p0) of (1.2) with initial value (0, p0), p0 ∈ J at time t = 0 is periodic in twith frequency Ω(p0) > 0.

(c) There is an integer r ≥ 0 such that for every 0 ≤ k ≤ r+7 the limit of ∂kp0Ω(p0) for p0 → Jl,Jrexists and does not vanish for k = 0. If 0 ∈ J then lim

p0→0∂p0Ω(p0) = 0.

GA 1.2. The real parts of the eigenvalues of A are all negative, bounded by a suitable constant c0 > 0:

ℜ(σ(A)) ≤ −c0.

Here and in what follows, σ(A) denotes the spectrum, i.e. the set of all eigenvalues of the matrixA. Moreover we assume that A is diagonalizable.

GA 1.3. Consider the Taylor expansion of order 3 in ε = 0 of the maps F and G, i.e. the representation

F (q, p, η, t, ε) =

3∑

j=1

εj F j(q, p, η, t) + ε4 F 4(q, p, η, t, ε)

G(q, p, t, ε) =3∑

j=1

εj Gj(q, p, t) + ε4G4(q, p, t, ε).

(1.3)

We assume that the maps F j , Gj , j = 1, 2, 3 in (1.3) admit a representation as finite Fourier series1

of degree N ∈ N with respect to t , i.e.

F j(q, p, η, t) =∑

|n|≤NF jn(q, p, η) e

int Gj(q, p, t) =∑

|n|≤NGjn(q, p) e

int. (1.4)

GA 1.4. The map F is affine with respect to η, i.e. ∂kηF (q, p, η, t, ε) = 0 for all k ≥ 2.

1Note that the functions Fjn, G

jn, n ∈ −N, . . . , N, j = 1, 2, 3 are complex valued functions. As system (1.1) is real, it

is easy to see that Fjn = F

j−n, G

jn = G

j−n, i.e. the complex conjugate valued functions.

1.1. The System under Consideration 3

1.1.3 A Short Overview on the Strategy Followed

The aim of this first chapter is to derive a plane (non–autonomous) system which asymptotically de-termines the qualitative behaviour of system (1.1). Considering (1.1) in the unperturbed case (ε = 0)we see that due to the stability of the matrix A (as assumed in GA 1.2), all solutions tend towards the(q, p)–plane η = 0. Hence the asymptotic behaviour in the unperturbed case is determined by the planeHamiltonian system (1.2).

In the case of a small perturbation (ε 6= 0 but small) we aim on a reduction to a plane system aswell. However, it will be necessary to consider different regions of the (q, p) phase space separately inorder to derive appropriate coordinates. Using invariant manifold theory we then show the existenceof an attractive two-dimensional (time–dependent) invariant manifold for the corresponding region andconsider the system restricted to this manifold. Following this way we yield a plane system, representingthe asymptotic behaviour in the corresponding domain for the perturbed case as well.

In sections 1.2–1.4 we deal with regions of periodic solutions of the Hamiltonian system (1.2) near anelliptic fixed point. (Note that the fixed point itself is not included in such a region). As the planeHamiltonian system admits an elliptic fixed point at (q, p) = (0, 0), there exists a periodic solution ofthe perturbed 2 + d–dimensional system (1.1) near the origin. This is dealed with in section 1.2. Sincethe stability of this periodic solution is essential for the asymptotic behaviour of (1.1), we will performchanges of coordinates in a way such that the region considered may be extended into this periodicsolution. This will be prepared in section 1.4.

Sections 1.5–1.6 deal with any region of periodic solutions of the Hamiltonian system. Introducing actionangle coordinates in section 1.5 it will be possible to establish the existence of an attractive invariantmanifold in section 1.6 and consider the ”restricted” plane system on the region chosen.

The entire process carried out in chapter 1 and chapter 2 will be presented in a form sufficiently explicitfor application on concrete examples. This requires more work in the theoretical part but on the otherhand leads to a form applicable in many situations. Moreover the author has tried to present the stepscarried out in a ”modular” manner, such that the results of certain sections may be applied independently.


1.2 The Periodic Solution

1.2.1 The Existence of a Unique Periodic Solution Near the Origin

As mentioned above, in this section we consider the case where the Hamiltonian system (1.2) admits anelliptic fixed point at the origin. The aim is to establish the existence of a unique 2π–periodic solutionof system (1.1) (for ε sufficiently small), located near the elliptic fixed point at the origin. This will becarried out by applying the following general result to our situation.

Lemma 1.2.1 Consider an ordinary differential equation of the form

x = εp (f(x) + g(x, t, ε)) , x ∈ Rm (1.5)

where p ∈ N and f(0) = 0. Let f and g be of class C r (r ≥ 1 or r = ω) and assume that g is T–periodic with respect to t and vanishes for ε = 0, i.e. g(x, t, 0) = 0 ∀x ∈ Rm, ∀t ∈ R. Moreover, letσ (Df(0)) ∩ i 2π

T Z = ∅ if p = 0 and detDf(0) 6= 0 if p > 0.

Then there exists an ε1 > 0 and a unique map x ∈ C r(R× (−ε1, ε1),Rm) such that x(t, 0) = 0 (∀t ∈ R)and for every |ε| < ε1, the mapping t 7→ x(t, ε) is a T–periodic solution of system (1.5).

PROOF: We prove this lemma in several steps.

1. First, define the map

g(x, t, ε) := f(x)−Df(0)x+ g(x, t, ε)

for x ∈ Rm, t ∈ R and ε ∈ R. Then we see that g is T –periodic with respect to t,

g(0, t, 0) = 0 and ∂xg(0, t, 0) = ∂xg(0, t, 0) = 0. (1.6)

Using this map g we may write (1.5) as follows:

x = εp (Df(0)x+ g(x, t, ε)) . (1.7)

Let x(t; t0, x0, ε) denote the solution of (1.5) with initial value x0 at time t0. By the uniqueness ofsolutions we have

x(t; t0, x0, ε) = x(t; t1, x(t1; t0, x0, ε), ε). (1.8)

Since g(x, t, ε) is T –periodic with respect to t, it follows that the flow induced by (1.5) is T –periodicas well, hence

x(t; t0, x0, ε) = x(t+ T ; t0 + T, x0, ε). (1.9)

As g vanishes for ε = 0 we finally note that x = 0 is a solution of (1.5) for ε = 0, hence

x(t; t0, 0, 0) = 0 ∀t, t0 ∈ R. (1.10)

1.2. The Periodic Solution 5

2. In a next step we shall establish the existence of a unique initial value ξ(ε) near x = 0 whichcorresponds to a T –periodic solution of (1.5). This will be shown by applying the Implicit FunctionTheorem to the following map:

R(ε, ξ) :=1

εp T

(

eεp T Df(0) − IR 2+d

)

ξ +1

T

∫ T

0

eεp (T−s)Df(0) g(x(s; 0, ξ, ε), s, ε) ds.

Under the conditions assumed, R ∈ C r(R× Rm,Rm). By (1.6) and (1.10) we find

R(0, ξ) =

1T

(

eT Df(0) − IR 2+d

)

ξ p = 0

Df(0) ξ p > 0

such that R(0, 0) = 0. Taking the partial derivative of R(0, ξ) we find

∂ξR(0, 0) =

1T

(

eT Df(0) − IR 2+d

)

p = 0

Df(0) p > 0.

Since by assumption σ (Df(0)) ∩ i 2πT Z = ∅ (if p = 0) and detDf(0) 6= 0 (if p > 0), we see that

det(∂ξR(0, 0)) =

1T det

(

eT Df(0) − IR 2+d

)

= 1T

∏

λ∈σ(Df(0))(eTλ − 1) 6= 0 p = 0

detDf(0) 6= 0 p > 0.

Hence it follows by the Implicit Function Theorem that there exists an ε1 > 0 as well as a uniquemap ξ ∈ C r((−ε1, ε1),Rm) with ξ(0) = 0 such that for every |ε| < ε1, R(ε, ξ(ε)) = 0.

In accordance with the representation (1.7) we write the solution x(t; 0, ξ0, ε) of (1.5) with initialvalue ξ0 at time t0 = 0 using the Variation of Constant formula, i.e.

x(t; 0, ξ0, ε) = eεp t Df(0) ξ0 + εp

∫ t

0

eεp (t−s)Df(0) g(x(s; 0, ξ0, ε), s, ε) ds.

By definition of R we thus find

R(ε, ξ) = 0 ⇔ εp T R(ε, ξ) = 0 ⇔ ξ = x(T ; 0, ξ, ε). (1.11)

Setting ξ = ξ(ε) therefore yields ξ(ε) = x(T ; 0, ξ(ε), ε).

3. It remains to show that for fixed |ε| < ε1 the initial value ξ(ε) generates a periodic solution of (1.5),indeed. We therefore define x(t, ε) := x(t; 0, ξ(ε), ε). Applying (1.8), (1.9) and (1.11) we find forany t ∈ R

x(t+ T, ε) = x(t+ T ; 0, ξ(ε), ε) = x(t+ T ;T, x(T ; 0, ξ(ε), ε), ε)

= x(t; 0, x(T ; 0, ξ(ε), ε), ε) = x(t; 0, ξ(ε), ε)

= x(t, ε).

Hence the solution x(t, ε) with initial value x(0, ε) = ξ(ε) is T –periodic .

Moreover, ξ(0) = 0 together with (1.10) imply

x(t, 0) = x(t; 0, ξ(0), 0) = x(t; 0, 0, 0) = 0.


4. With the help of the statements given by the Implicit Function Theorem on the uniqueness, rangeand domain of the map ξ, it eventually may be shown that the map t 7→ x(t, ε) is the only T –periodicsolution close to the origin satisfying x(t, 0) = 0.

Therefore the statement given in lemma 1.2.1 is proved.

It now is a simple consequence of the preceeding lemma that system (1.1) admits a unique 2π–periodicsolution (q, p, η) close to the origin. This is carried out in the following lemma.

Lemma 1.2.2 There exists ε1 > 0 as well as a unique map (q, p, η) ∈ Cω(R × (−ε1, ε1),R 2+d) suchthat for fixed |ε| < ε1 the map t 7→ (q, p, η)(t, ε) is a 2π–periodic solution of (1.1) and for ε = 0,(q, p, η)(t, 0) = 0 ∀t ∈ R.

For simplicity we will omit the parameter a in the notation (q, p, η) unless needed explicitely.

PROOF: For x = (q, p, η) ∈ R 2+d we set

f(x) := f(q, p, η) :=

(

J∇H(q, p)Aη

)

g(x, t, ε) := g(q, p, η, t, ε) :=

(

F (q, p, η, t, ε)G(q, p, t, ε)

)

.

By assumption GA 1.1a we have f(0) =

(

J∇H(0, 0)A 0

)

= 0 and

σ(Df(0)) = σ

([

JD2H(0, 0) 00 A

])

= σ(JD2H(0, 0)) ∪ σ(A), (1.12)

such that from σ(

JD2H(0, 0))

=

± i√

∂2qH(0, 0) ∂2pH(0, 0)

and GA 1.1a together with GA 1.2 we

deduce σ (Df(0)) ∩ iZ = ∅.

Taking into account the assumptions made in section 1.1.1 for F , G and H it is readily seen that we arein the position to apply lemma 1.2.1 (where m = 2 + d, p = 0, r = ω and T = 2π). Hence the proof oflemma 1.2.2 is a consequence of lemma 1.2.1.


1.2.2 The Transformation into the Periodic Solution

The purpose of this section is to transform the coordinates of system (1.1) in a way, such that the originbecomes a fixed point. This may be done by performing a (time–dependent) translation into the periodicsolution (q, p, η).

More precisely we will use the Taylor / Fourier expansions (1.3), (1.4) assumed in GA 1.3 to explicitelycalculate a similar representation of the corresponding vector field in the new coordinates. This will beprepared in the following lemma:

Lemma 1.2.3 Consider a linear inhomogenous differential equation on R 2+d of the following type

x = B x+∑

|n|≤Nbne

int, (1.13)

where bn ∈ C 2+d for every |n| ≤ N , bn = b−n and σ (B)∩iZ = ∅. Then there exists a unique 2π–periodicsolution given by

x(t) =∑

|n|≤N[i n IC 2+d −B]−1 bn e

int. (1.14)

PROOF: Note first that since σ (B)∩ iZ = ∅, the inverse of the matrix i n IC 2+d −B exists. It is evidentthat the function x presented in (1.14) is 2π–periodic with respect to t. Moreover

x(t)−B x(t) =∑

|n|≤Ni n [i n IC 2+d −B]

−1bn e

int −B∑

|n|≤N[i n IC 2+d −B]

−1bn e

int

=∑

|n|≤N[i n IC 2+d −B] [i n IC 2+d −B]

−1bn e

int

=∑

|n|≤Nbn e

int,

such that x is a solution of (1.13), indeed.

Consider any further 2π–periodic solution y of (1.13). Writing its Fourier expansion y =∑

n∈N

cn eint and

calculating y−B y one then compares the result with∑

|n|≤Nbn e

int which implies cn = [i n IC 2+d −B]−1

bn

and thus x = y. Hence x is unique as claimed.

We now are in the position to prove the main result of this section.

Proposition 1.2.4 Let (q, p, η) denote the 2π–periodic solution of system (1.1) for |ε| < ε1, asserted inlemma 1.2.2 and perform the following change of coordinates in the (q, p, η, t, ε)–space:

(q, p, η, t, ε) = ((q, p, η)(t, ε), 0, 0) + (Q, P ,H, t, ε), (1.15)


where2 (Q, P ) ∈ R2, H ∈ Rd, t ∈ R and |ε| < ε1. Then (1.1) transforms into the system

( ˙Q, ˙P ) = J∇H(Q, P ) + F (Q, P ,H, t, ε)

H = AH+ G(Q, P , t, ε),(1.16)

where the following statements hold :

• The mappings F and G are of class Cω, vanish at the origin (Q, P ,H) = 0 and admit the repre-sentation3

F (Q, P ,H, t, ε) =

3∑

j=1

εj F j(Q, P ,H, t) + ε4 F 4(Q, P ,H, t, ε)

G(Q, P , t, ε) =

3∑

j=1

εjGj(Q, P , t) + ε4 G4(Q, P , t, ε)

(1.17)

where F j and Gj , (j = 1, . . . , 4) are 2π–periodic with respect to t.

• The map H 7→ F (Q, P ,H, t, ε) is affine.

• The mappings F 1, F 2, G1 and G2 may be expressed in terms of the original vector field of system(1.1):

F 1(Q, P ,H, t) = F 1(Q, P ,H, t)−∑

|n|≤N∆(n, Q, P )F 1

n(0, 0, 0) eint

F 2(Q, P ,H, t) = F 2(Q, P ,H, t)−∑

|n|≤N∆(n, Q, P )F 2

n(0, 0, 0)eint

+∑

|n|,|n|≤N

[

12JD

3H(Q, P )(

α1,1n,1, α

1,1n,1

)

+[

∂(q,p)F1n(Q, P ,H)−∆(n+ n, Q, P ) ∂(q,p)F

1n(0, 0, 0)

]

α1,1n,1

+[

∂ηF1n(Q, P ,H)−∆(n+ n, Q, P ) ∂ηF

1n(0, 0, 0)

]

α1,1n,2

]

ei(n+n)t

G1(Q, P , t) = G1(Q, P , t)−G1(0, 0, t)

G2(Q, P , t) = G2(Q, P , t)−G2(0, 0, t) +∑

|n|,|n|≤N

[

∂(q,p)G1n(Q, P )− ∂(q,p)G

1n(0, 0)

]

α1,1n,1

(1.18)

where

∆(n, Q, P ) := [i n IC 2 − JD2H(Q, P )][

i n IC 2 − JD2H(0, 0)]−1

α1,1n,1 =

[

i n IC 2 − JD2H(0, 0)]−1

F 1n(0, 0, 0)

α1,1n,2 = [i n IC d −A]

−1G1n(0, 0).

(1.19)

2The letter H must be read as ”upper eta”3for the application in chapter 4 it suffices to consider the expansions including terms of order O(ε2) of F and of order

O(ε) of G. The formulae for O(ε3)–terms are provided in order to enable a more detailed discussion on the capture inresonance, cf. section 2.3.5.


• Moreover, F 1, F 2, G1 and G2 may be represented as Fourier polynomials in t, similar to therepresentation (1.4), i.e.

F j(Q, P ,H, t) =∑

|n|≤jNF jn(Q, P ,H, t) e

int Gj(Q, P , t) =∑

|n|≤jNGjn(Q, P , t) e

int (1.20)

• The values of the map F 3 may be expressed in an analogous way:

F 3(Q, P ,H, t) =F 3(Q, P ,H, t)−∑

|n|≤N∆(n, Q, P )F 3

n(0, 0, 0)eint

+∑

|n|,|n|≤N

[

JD3H(Q, P )(

α1,1n,1, α

2,1n,1

)

+[

∂(q,p)F2n(Q, P ,H)−∆(n+ n, Q, P ) ∂(q,p)F

2n(0, 0, 0)

]

α1,1n,1

+[

∂ηF2n(Q, P ,H)−∆(n+ n, Q, P ) ∂ηF

2n(0, 0, 0)

]

α1,1n,1

+[

∂(q,p)F1n(Q, P ,H)−∆(n+ n, Q, P ) ∂(q,p)F

1n(0, 0, 0)

]

α2,1n,1

+[

∂ηF1n(Q, P ,H)−∆(n+ n, Q, P ) ∂ηF

1n(0, 0, 0)

]

α2,1n,1

]

ei(n+n)t

+∑

|n|,|n|,|n|≤N

[

JD3H(Q, P )(

α1,1n,1, α

2,2n,n,1

)

+ 16

(

JD4H(Q, P )−∆(n+ n+ n, Q, P )JD4H(0))

(

α1,1n,1

)(

α1,1n,1

)(

α1,1n,1

)

+ 12

(

∂2(q,p)F1n(Q, P ,H)−∆(n+ n+ n, Q, P ) ∂2(q,p)F

1n(0, 0, 0)

)(

α1,1n,1, α

1,1n,1

)

+ 12

(

∂η∂(q,p)F1n(Q, P ,H)−∆(n+ n+ n, Q, P ) ∂η∂(q,p)F

1n(0, 0, 0)

)

(

α1,1n,1, α

1,1n,2

)

+ 12

(

∂(q,p)∂ηF1n(Q, P ,H)−∆(n+ n+ n, Q, P ) ∂(q,p)∂ηF

1n(0, 0, 0)

)

(

α1,1n,2, α

1,1n,1

)

+[

∂(q,p)F1n(Q, P ,H)−∆(n+ n, Q, P ) ∂(q,p)F

1n(0, 0, 0)

]

α2,2n,n,1

+[

∂ηF1n(Q, P ,H)−∆(n+ n, Q, P ) ∂ηF

1n(0, 0, 0)

]

α2,2n,n,1

]

ei(n+n+n)t.

(1.21)

where in addition

α2,1n,1 =

[

i n IC 2 − JD2H(0, 0)]−1

F 2n(0, 0, 0) α2,1

n,2 = [i n IC d −A]−1

G2n(0, 0)

α2,2n,n,1 =

[

i (n+ n) IC 2 − JD2H(0, 0)]−1

(

∂(q,p)F1n(0, 0, 0)α

1,1n,1 + ∂ηF

1n(0, 0, 0)α

1,1n,2

)

.

PROOF: In order to simplify the notation, we use the same abbreviations as introduced in the proof oflemma 1.2.2:

x := (q, p, η) x(t, ε) :=(q, p, η)(t, ε) y := (Q, P ,H)

f(x) :=

(

J∇H(q, p)Aη

)

g(x, t, ε) :=

(

F (q, p, η, t, ε)G(q, p, t, ε)

) (1.22)

such that system (1.1) defined for x ∈ R 2+d reads

x = f(x) + g(x, t, ε). (1.23)


In view of assumption GA 1.3 we rewrite g(x, t, ε) as follows:

g(x, t, ε) =

3∑

j=1

εj gj(x, t) + ε4 g4(x, t, ε) =

3∑

j=1

εj∑

|n|≤Ngjn(x) e

int + ε4 g4(x, t, ε) (1.24)

where we have set

gjn(x) =

(

F jn(q, p, η)Gjn(q, p)

)

∈ Cω(R 2+d,C 2+d), |n| ≤ N, j = 1, 2, 3

and

g4(x, t, ε) =

(

F 4(q, p, η, t, ε)G4(q, p, t, ε)

)

∈ Cω(R 2+d × R× R,R 2+d).

The transformation defined in (1.15) corresponds to the time–dependent, 2π–periodic translation

x = x(t, ε) + y

defined for t ∈ R and |ε| < ε1. Expressing (1.23) in the new coordinates yields

y = f(x(t, ε) + y) + g(x(t, ε) + y, t, ε)− ˙x(t, ε) =: f(y, t, ε). (1.25)

Note that f ∈ Cω(R 2+d × R× (−ε1, ε1),R 2+d) as f , g and x are of class Cω. As x(t, ε) is a solution of(1.1) and hence of (1.23) as well, we find

f(0, t, ε) = 0. (1.26)

Moreover, as x and g vanish for ε = 0, it follows at once that

f(y, t, 0) = f(y). (1.27)

Since the last d components of

f(y, t, ε)− f(y, t, 0) =

(

J ∇H(q + Q, p+ P )A (η + H)

)

+

(

F (q + Q, p+ P , η + H, t, ε)G(q + Q, p+ P , t, ε)

)

−(

J ∇H(Q, P )AH

)

− ˙x(t, ε)

=

(

J ∇H(q + Q, p+ P )− J ∇H(Q, P )− J∇H(q, p)A η

)

+

(

F (q + Q, p+ P , η + H, t, ε)− F (q, p, η, t, ε)G(q + Q, p+ P , t, ε)−G(q, p, t, ε)

)

(1.28)

do not depend on H, we may split this vector f(y, t, ε) − f(y, t, 0) into a two–dimensional componentwhich depends on H and a second, d–dimensional component, independent of H. More precisely we arein the position to introduce the following abbreviations :

(

F (Q, P ,H, t, ε)G(Q, P , t, ε)

)

:= f(y, t, ε)− f(y, t, 0)

We continue with the following steps:


1. Consider the Taylor expansion of f at ε = 0, i.e. the representation

f(y, t, ε) = f(y, t, 0) + ε ∂εf(y, t, 0) +12ε

2∂2ε f(y, t, 0) +16ε

3∂3ε f(y, t, 0) + ε4f4(y, t, ε) (1.29)

where f4(y, t, ε) is of class Cω(R 2+d × R × (−ε1, ε1),R 2+d) and 2π–periodic with respect to t.Setting

(

F j(Q, P ,H, t)Gj(Q, P , t)

)

:= 1j!∂

jε f(y, t, 0) j = 1, 2, 3

(

F 4(Q, P ,H, t, ε)G4(Q, P , t, ε)

)

:= f4(y, t, ε),

(1.30)

and taking into account that (1.26), (1.28) imply(

F (0, 0, 0, t, ε)G(0, 0, t, ε)

)

= −f(0) = 0

we find the first statement claimed to be proved at once.

2. In order to prove the second statement we note that by (1.28), (1.30)

F (Q, P ,H, t, ε) = J ∇H(q + Q, p+ P )− J ∇H(Q, P )− J∇H(q, p)

+F (q + Q, p+ P , η + H, t, ε)− F (q, p, η, t, ε)

such that the affinity of F assumed in GA 1.4 implies the affinity of F (with respect to H).

3. We determine the Taylor coefficients in (1.29).

Using (1.26) we have

∂jε f(0, t, 0) = 0 j = 1, 2, 3. (1.31)

On the other hand, from definition (1.25) we derive

∂εf(y, t, ε) =Df(x(t, ε) + y) ∂εx(t, ε) + ∂xg(x(t, ε) + y, t, ε) ∂εx(t, ε)

+ ∂εg(x(t, ε) + y, t, ε)− ∂ε ˙x(t, ε),(1.32)

∂2ε f(y, t, ε) =D2f(x(t, ε) + y) ∂εx(t, ε)[2] +Df(x(t, ε) + y) ∂2ε x(t, ε)

+ ∂2xg(x(t, ε) + y, t, ε) ∂εx(t, ε)[2] + ∂xg(x(t, ε) + y, t, ε) ∂2ε x(t, ε)

+ 2 ∂ε∂xg(x(t, ε) + y, t, ε) ∂εx(t, ε) + ∂2εg(x(t, ε) + y, t, ε)

− ∂2ε ˙x(t, ε)

(1.33)

∂3ε f(y, t, ε) =D3f(x(t, ε) + y) ∂εx(t, ε)[3]

+ 3D2f(x(t, ε) + y) (∂εx(t, ε), ∂2ε x(t, ε)) +Df(x(t, ε) + y) ∂3ε x(t, ε)

+ ∂3xg(x(t, ε) + y, t, ε) ∂εx(t, ε)[3]

+ 3 ∂2xg(x(t, ε) + y, t, ε) (∂εx(t, ε), ∂2ε x(t, ε)) + ∂xg(x(t, ε) + y, t, ε) ∂3ε x(t, ε)

+ 3 ∂ε∂2xg(x(t, ε) + y, t, ε) ∂εx(t, ε)

[2]

+ 3 ∂2ε∂xg(x(t, ε) + y, t, ε) ∂εx(t, ε)

+ 3 ∂ε∂xg(x(t, ε) + y, t, ε) ∂2ε x(t, ε) + ∂3εg(x(t, ε) + y, t, ε)

− ∂3ε ˙x(t, ε)

(1.34)


where the notation v[j] must be understood as applying the corresponding multilinear–form on thej vectors (v, . . . , v). Taking into account that by (1.24)

∂jxg(y, t, 0) = 0 j = 1, 2, 3 ∂ε∂jxg(y, t, 0) = ∂jxg

1(y, t) j = 0, 1, 2

∂2ε∂jxg(y, t, 0) = 2 ∂jxg

2(y, t) j = 0, 1 ∂3εg(y, t, 0) = 6 g3(y, t),

we therefore see that setting ε = 0, (1.32), (1.33) and (1.34) reduce to

∂εf(y, t, 0) =Df(y) ∂εx(t, 0) + g1(y, t)− ∂t∂εx(t, 0)

∂2ε f(y, t, 0) =D2f(y) ∂εx(t, 0)[2] +Df(y) ∂2ε x(t, 0)

+ 2 ∂xg1(y, t) ∂εx(t, 0) + 2 g2(y, t)− ∂t∂

2ε x(t, 0)

∂3ε f(y, t, 0) =D3f(y) ∂εx(t, 0)[3]

+ 6D2f(y) (∂εx(t, 0),12∂

2ε x(t, 0)) +Df(y) ∂3ε x(t, 0)

+ 3 ∂2xg1(y, t) ∂εx(t, 0)

[2] + 6 ∂xg2(y, t) ∂εx(t, 0)

+ 6 ∂xg1(y, t) 1

2∂2ε x(t, 0) + 6 g3(y, t)

− ∂t∂3ε x(t, 0).

(1.35)

4. In a next step we compute the functions ∂εx(t, 0), ∂2ε x(t, 0) and ∂3ε x(t, 0) by solving differential

equations :

Recall that by GA 1.1a D3H(0, 0) = 0 such that by definition of f , D2f(0) = 0. Therefore (1.24)together with (1.31), (1.35) yields the following linear inhomogeneous differential equations

∂t∂εx(t, 0) = Df(0) ∂εx(t, 0) + g1(0, t) = Df(0) ∂εx(t, 0) +∑

|n|≤Ng1n(0) e

int, (1.36)

∂t∂2ε x(t, 0) = Df(0) ∂2ε x(t, 0) + 2 ∂xg

1(0, t) ∂εx(t, 0) + 2 g2(0, t) (1.37)

and

∂t∂3ε x(t, 0) =Df(0) ∂3ε x(t, 0) +D3f(0) ∂εx(t, 0)

[3]

+ 3 ∂2xg1(0, t) ∂εx(t, 0)

[2] + 6 ∂xg2(0, t) ∂εx(t, 0)

+ 6 ∂xg1(0, t) 1

2∂2ε x(t, 0) + 6 g3(0, t).

(1.38)

As we have shown in (1.12) in the proof of lemma 1.2.2, σ (Df(0)) ∩ iZ = ∅. Hence lemma 1.2.3may be applied to equation (1.36). Therefore the unique 2π–periodic solution ∂εx(t, 0) of (1.36)is given by

∂εx(t, 0) =∑

|n|≤N~α1,1n eint, where ~α1,1

n := [i n IC 2+d −Df(0)]−1

g1n(0). (1.39)

Let us rewrite the differential equation (1.37) using (1.24) and (1.39):

∂t∂2ε x(t, 0) = Df(0) ∂2ε x(t, 0) + 2

∑

|n|≤NDg1n(0)e

int

∑

|n|≤N~α1,1n eint

+ 2∑

|n|≤Ng2n(0) e

int

= Df(0) ∂2ε x(t, 0) + 2∑

|n|≤Ng2n(0) e

int + 2∑

|n|,|n|≤NDg1n(0) ~α

1,1n ei(n+n)t.


Solving this equation with the help of lemma 1.2.3 again we obtain

12∂

2ε x(t, 0) =

∑

|n|≤N~α2,1n eint +

∑

|n|,|n|≤N~α2,2n,n e

i(n+n)t,

with ~α2,1n := [i n IC 2+d −Df(0)]

−1g2n(0)

~α2,2n,n := [i (n+ n) IC 2+d −Df(0)]

−1Dg1n(0) ~α

1,1n .

(1.40)

Finally we proceed in an analogous way to obtain

16∂

3ε x(t, 0) =

∑

|n|,|n|,|n|≤N~α3,3n,n,n e

i(n+n+n)t +∑

|n|,|n|≤N~α3,2n,n e

i(n+n)t +∑

|n|≤N~α3,1n eint (1.41)

where

~α3,3n,n,n = [i (n+ n+ n) IC 2+d −Df(0)]

−1

(

16D

3f(0)(~α1,1n , ~α1,1

n , ~α1,1n )

+ 12D

2g1n(0)(~α1,1n , ~α1,1

n ) +Dg1n(0) ~α2,2n,n

)

~α3,2n,n = [i (n+ n) IC 2+d −Df(0)]−1

(

Dg1n(0)~α2,1n +Dg2n(0)~α

1,1n

)

~α3,1n = [i n IC 2+d −Df(0)]−1 g3n(0).

(1.42)

5. In order to gain expressions for the coefficient maps ∂εf(y, t, 0),12∂

2ε f(y, t, 0) and 1

6∂3ε f(y, t, 0) in

terms of known quantities, we combine the results derived in the first two steps. Let us introducethe notations

∆(n, Q, P ) := [i n IC 2 − JD2H(Q, P )][

i n IC 2 − JD2H(0, 0)]−1

M(n, Q, P ) :=

[

∆(n, Q, P ) 00 IC d

]

= [i n IC 2+d −Df(y)] [i n IC 2+d −Df(0)]−1.

(1.43)

Note that ∆(n, 0, 0) = IC 2 andM(n, 0, 0) = IC 2+d . Using the identities (1.24) and (1.39) we rewritethe first equation in (1.35):

∂εf(y, t, 0) =∑

|n|≤N

(

Df(y) ~α1,1n + g1n(y)− i n ~α1,1

n

)

eint

=∑

|n|≤N

(

g1n(y)− [i n IC 2+d −Df(y)] ~α1,1n

)

eint

=∑

|n|≤N

(

g1n(y)−M(n, Q, P ) g1n(0))

eint. (1.44)

The analogous result for 12∂

2ε f(y, t, 0) is achieved by substituting (1.24), (1.40) into the second


equation of (1.35):

12∂

2ε f(y, t, 0) = 1

2D2f(y)

∑

|n|≤N~α1,1n eint,

∑

|n|≤N~α1,1n eint

+∑

|n|≤NDf(y) ~α2,1

n eint +∑

|n|,|n|≤NDf(y) ~α2,2

n,n ei(n+n)t

+

∑

|n|≤NDg1n(y) e

int

∑

|n|≤N~α1,1n eint

+∑

|n|≤Ng2n(y)e

int

−∑

|n|≤Ni n ~α2,1

n eint −∑

|n|,|n|≤Ni (n+ n) ~α2,2

n,n ei(n+n)t

=∑

|n|≤N

[

Df(y) ~α2,1n + g2n(y)− i n ~α2,1

n

]

eint

+∑

|n|,|n|≤N

[

12D

2f(y)(

~α1,1n , ~α1,1

n

)

+Df(y) ~α2,2n,n +Dg1n(y) ~α

1,1n

−i (n+ n) ~α2,2n,n

]

ei(n+n)t.

Using the abbreviations defined in (1.43) together with the definitions of ~α2,1n , ~α2,2

n,n given in (1.40)we find

12∂

2ε f(y, t, 0) =

∑

|n|≤N

[

g2n(y)− [i n IC 2+d −Df(y)] ~α2,1n

]

eint

+∑

|n|,|n|≤N

[

12D

2f(y)(

~α1,1n , ~α1,1

n

)

+Dg1n(y) ~α1,1n

− [i (n+ n) IC 2+d −Df(y)] ~α2,2n,n

]

ei(n+n)t

=∑

|n|≤N

[

g2n(y)− [i n IC 2+d −Df(y)] [i n IC 2+d −Df(0)]−1

g2n(0)]

eint

+∑

|n|,|n|≤N

[

12D

2f(y)(

~α1,1n , ~α1,1

n

)

+Dg1n(y) ~α1,1n

− [i (n+ n) IC 2+d −Df(y)] [i (n+ n) IC 2+d −Df(0)]−1

Dg1n(0) ~α1,1n

]

ei(n+n)t

hence

12∂

2ε f(y, t, 0) =

∑

|n|≤N

[

g2n(y)−M(n, Q, P ) g2n(0)]

eint

+∑

|n|,|n|≤N

[

12D

2f(y)(

~α1,1n , ~α1,1

n

)

+[

Dg1n(y)−M(n+ n, Q, P )Dg1n(0)]

~α1,1n

]

ei(n+n)t. (1.45)

In a similar way we deduce the following representation of 16∂

3ε f(y, t, 0) from (1.24), (1.41) and the


last equation in (1.35)

16∂

3ε f(y, t, 0) = 1

6D3f(y)

(

∑


∑


∑

|n|≤N~α1,1n eint

)

+D2f(y)

(

∑


∑

|n|≤N~α2,1n eint

)

+D2f(y)

(

∑


∑

|n|,|n|≤N~α2,2n,n e

i(n+n)t

)

+Df(y)

(

∑

|n|,|n|,|n|≤N~α3,3n,n,n e

i(n+n+n)t

+∑

|n|,|n|≤N~α3,2n,n e

i(n+n)t +∑

|n|≤N~α3,1n eint

)

+ 12

(

∑

|n|≤ND2g1n(y)e

int

)(

∑


∑

|n|≤N~α1,1n eint

)

+

(

∑

|n|≤NDg2n(y)e

int

)(

∑

|n|≤N~α1,1n eint

)

+

(

∑

|n|≤NDg1n(y)e

int

)(

∑

|n|≤N~α2,1n eint +

∑

|n|,|n|≤N~α2,2n,n e

i(n+n)t

)

+∑

|n|≤Ng3n(y)e

int

−∑

|n|,|n|,|n|≤Ni (n+ n+ n) ~α3,3

n,n,n ei(n+n+n)t

−∑

|n|,|n|≤Ni (n+ n) ~α3,2

n,n ei(n+n)t

−∑

|n|≤Ni n ~α3,1

n eint,

thus

16∂

3ε f(y, t, 0) =

∑

|n|≤N

[

Df(y)~α3,1n + g3n(y)− i n ~α3,1

n

]

eint

+∑

|n|,|n|≤N

[

D2f(y)(

~α1,1n , ~α2,1

n

)

+Df(y)~α3,2n,n +Dg2n(y)~α

1,1n

+Dg1n(y)~α2,1n − i (n+ n) ~α3,2

n,n

]

ei(n+n)t

+∑

|n|,|n|,|n|≤N

[

16D

3f(y)(

~α1,1n , ~α1,1

n , ~α1,1n

)

+D2f(y)(

~α1,1n , ~α2,2

n,n

)

+Df(y)~α3,3n,n,n + 1

2D2g1n(y)

(

~α1,1n , ~α1,1

n

)

+Dg1n(y)~α2,2n,n − i (n+ n+ n)~α3,3

n,n,n

]

ei(n+n+n)t


which by (1.40), (1.42) eventually leads to

16∂

3ε f(y, t, 0) =

∑

|n|≤N

[

g3n(y)−M(n, Q, P ) g3n(0)

]

eint

+∑

|n|,|n|≤N

[

D2f(y)(

~α1,1n , ~α2,1

n

)

+[


~α1,1n

+[


~α2,1n

]

ei(n+n)t

+∑

|n|,|n|,|n|≤N

[

D2f(y)(

~α1,1n , ~α2,2

n,n

)

+ 16

[

D3f(y)−M(n+ n+ n, Q, P )D3f(0)]

(

~α1,1n , ~α1,1

n , ~α1,1n

)

+ 12

[

D2g1n(y)−M(n+ n+ n, Q, P )D2g1n(0)]

(

~α1,1n , ~α1,1

n

)

+[

Dg1n(y)−M(n+ n+ n, Q, P )Dg1n(0)]

~α2,2n,n)

]

ei(n+n+n)t.

(1.46)

6. In a next step, we split the quantities ∂εf(y, t, 0),12∂

2ε f(y, t, 0) and 1

6∂3ε f(y, t, 0) into two compo-

nents, expressed in terms of the maps F jn and Gjn. This will lead us to the formulae claimed in(1.18) and (1.21).

Using definitions (1.43), (1.24) we rewrite (1.44) as follows :

∂εf(y, t, 0) =∑

|n|≤N

(

F 1n(Q, P ,H)−∆(n, Q, P )F 1

n(0, 0, 0)G1n(Q, P )−G1

n(0, 0)

)

eint. (1.47)

For convenience we split the vectors ~αj,1n , ~α2,2n,n into two components of dimension 2 and d :

~αj,1n =:

(

αj,1n,1αj,1n,2

)

~α2,2n,n =:

(

α2,2n,n,1

α2,2n,n,2

)

By definition (1.22) we find derivatives of f to be diagonal operators in the following sense :

Df(y) ∼=[

JD2H(Q, P ) 00 A

]

D2f(y)

(

(Q1, P1)H1

)(

(Q2, P2)H2

)

=

(

JD3H(Q, P )(Q1, P1)(Q2, P2)0

)

(1.48)

D3f(y)

(

(Q1, P1)H1

)(

(Q2, P2)H2

)(

(Q3, P3)H3

)

=

(

JD4H(Q, P )(Q1, P1)(Q2, P2)(Q3, P3)0

)

.

Note that by simple consequence,

[i n IC 2+d −Df(y)] =

[

i n IC 2 − JD2H(Q, P ) 00 i n ICd −A

]

.

Together with the representation of ~α1,1n introduced above, we obtain

D2f(y)(

~α1,1n , ~αk,jn

)

=

(

JD3H(Q, P )(

α1,1n,1, α

k,jn,1

)

0

)

k, j = 1, 2, (1.49)


and as Gjn does not depend on η, we have

Dgjn(y) =

[

∂(q,p)Fjn(Q, P ,H) ∂ηF

jn(Q, P ,H)

∂(q,p)Gjn(Q, P ) 0

]

. (1.50)

Hence equation (1.45) reads

12∂

2ε f(y, t, 0) =

∑

|n|≤N

[(

F 2n(Q, P ,H)G2n(Q, P )

)

−(

∆(n, Q, P )F 2n(0, 0, 0)

G2n(0, 0)

)]

eint

+∑

|n|,|n|≤N

[

12

(

JD3H(Q, P )(

α1,1n,1, α

1,1n,1

)

0

)

(1.51)

+

(

[

∂(q,p)F1n(Q, P ,H)−∆(n+ n, Q, P ) ∂(q,p)F

1n(0, 0, 0)

]

α1,1n,1

[

∂(q,p)G1n(Q, P )− ∂(q,p)G

1n(0, 0)

]

α1,1n,1

)

+

([

∂ηF1n(Q, P ,H)−∆(n+ n, Q, P ) ∂ηF

1n(0, 0, 0)

]

α1,1n,2

0

)]

ei(n+n)t.

We finally calculate the corresponding representation for 16∂

3ε f(y, t, 0). Since F is affine with respect

to η (GA 1.4) we have ∂2ηF1n(Q, P ,H) = 0 such that

D2g1n(y)(~α1,1n , ~α1,1

n ) =

∂2(q,p)F1n(Q, P ,H)

(

α1,1n,1, α

1,1n,1

)

∂2(q,p)G1n(Q, P )

(

α1,1n,1, α

1,1n,1

)

(1.52)

+

(

∂η∂(q,p)F1n(Q, P ,H)

(

α1,1n,1, α

1,1n,2

)

+ ∂(q,p)∂ηF1n(Q, P ,H)

(

α1,1n,2, α

1,1n,1

)

0

)

and considering (6) we find

D3f(y)(~α1,1n , ~α1,1

n , ~α1,1n ) =

(

JD4H(Q, P )(

α1,1n,1

)(

α1,1n,1

)(

α1,1n,1

)

0

)

. (1.53)

Applying (1.49)–(1.53) on (1.46) then yields

16∂

3ε f(y, t, 0) =

∑

|n|≤N

(

F 3n(Q, P ,H)−∆(n, Q, P )F 3

n(0, 0, 0)G3n(Q, P )−G3

n(0, 0)

)

eint

+∑

|n|,|n|≤N

[

(

JD3H(Q, P )(

α1,1n,1, α

2,1n,1

)

0

)

+

(

[

∂(q,p)F2n(Q, P ,H)−∆(n+ n, Q, P ) ∂(q,p)F

2n(0, 0, 0)

]

α1,1n,1

[

∂(q,p)G2n(Q, P )− ∂(q,p)G

2n(0, 0)

]

α1,1n,2

)

+

([

∂ηF2n(Q, P ,H)−∆(n+ n, Q, P ) ∂ηF

2n(0, 0, 0)

]

α1,1n,1

0

)

+

(

[

∂(q,p)F1n(Q, P ,H)−∆(n+ n, Q, P ) ∂(q,p)F

1n(0, 0, 0)

]

α2,1n,1

[

∂(q,p)G1n(Q, P )− ∂(q,p)G

1n(0, 0)

]

α2,1n,2

)

+

([

∂ηF1n(Q, P ,H)−∆(n+ n, Q, P ) ∂ηF

1n(0, 0, 0)

]

α2,1n,1

0

)]

ei(n+n)t

(1.54)


+∑

|n|,|n|,|n|≤N

[

(

JD3H(Q, P )(

α1,1n,1, α

2,2n,n,1

)

0

)

+ 16

(

(

JD4H(Q, P )−∆(n+ n+ n, Q, P )JD4H(0))

(

α1,1n,1

)(

α1,1n,1

)(

α1,1n,1

)

0

)

+ 12

(

∂2(q,p)F1n(Q, P ,H)−∆(n+ n+ n, Q, P ) ∂2(q,p)F

1n(0, 0, 0)

)(

α1,1n,1, α

1,1n,1

)

(

∂2(q,p)G1n(Q, P )− ∂2(q,p)G

1n(0, 0)

)(

α1,1n,1, α

1,1n,1

)

+ 12

(

(

∂η∂(q,p)F1n(Q, P ,H)−∆(n+ n+ n, Q, P ) ∂η∂(q,p)F

1n(0, 0, 0)

)

(

α1,1n,1, α

1,1n,2

)

0

)

+ 12

(

(

∂(q,p)∂ηF1n(Q, P ,H)−∆(n+ n+ n, Q, P ) ∂(q,p)∂ηF

1n(0, 0, 0)

)

(

α1,1n,2, α

1,1n,1

)

0

)

+

(

[

∂(q,p)F1n(Q, P ,H)−∆(n+ n, Q, P ) ∂(q,p)F

1n(0, 0, 0)

]

α2,2n,n,1

[

∂(q,p)G1n(Q, P )− ∂(q,p)G

1n(0, 0)

]

α2,2n,n,2

)

+

([

∂ηF1n(Q, P ,H)−∆(n+ n, Q, P ) ∂ηF

1n(0, 0, 0)

]

α2,2n,n,1

0

)]

ei(n+n+n)t.

7. Summarizing the identities given in (1.25), (1.29) and (1.27) we consider the transformed system

y = f(y, t, ε) = f(y) + ε∂εf(y, t, 0) +12ε

2∂2ε f(y, t, 0) +16ε

3∂3ε f(y, t, 0) + ε4f4(y, t, ε)

which by (1.22), (1.30) may be represented in the form

(

( ˙Q, ˙P )

H

)

=

(

J∇H(Q, P )AH

)

+

3∑

j=1

εj(

F j(Q, P ,H, t)Gj(Q, P , t)

)

+ ε4(

F 4(Q, P ,H, t, ε)G4(Q, P , t, ε)

)

.

Thus the identities (1.18) hold as it has been established in (1.47), (1.51) respectively.

8. In order to obtain the formula given in (1.21) one has to consider the first two components of16∂

3ε f(y, t, 0), which by (1.30) represents the vector–valued map F 3(Q, P ,H, t).

9. It remains to prove the formulae given for the quantities α1,1n,1, α

1,1n,2 etc. :

From the definition of (1.24) of gjn(y) we have

gjn(0) =

(

F jn(0, 0, 0)Gjn(0, 0)

)

j = 1, 2, 3

hence, by definitions (1.39), (1.40) of the vectors ~αj,1n ,

~αj,1n =

(

[

i n IC 2 − JD2H(0, 0)]−1

F jn(0, 0, 0)

[i n IC d −A]−1

Gjn(0, 0)

)

j = 1, 2.

Together with (1.50) this implies

Dg1n(y) ~α1,1n =

(

∂(q,p)F1n(Q, P ,H)α

1,1n,1 + ∂ηF

1n(Q, P ,H)α

1,1n,2

∂(q,p)G1n(Q, P )α

1,1n,1

)


such that definition (1.40) reads

~α2,2n,n =

(

[

i (n+ n) IC 2 − JD2H(0, 0)]−1

(

∂(q,p)F1n(0, 0, 0)α

1,1n,1 + ∂ηF

1n(0, 0, 0)α

1,1n,2

)

[i (n+ n) IC d −A]−1

∂(q,p)G1n(0, 0)α

1,1n,1

)

.

We therefore have established all assertions made in proposition 1.2.4.


1.3 Some Illustrative Examples

As explained in section 1.1.3, the strategy of this chapter consists in proving the existence of a local,attractive, two–dimensional invariant manifold. Once this step has been accomplished the qualitativediscussion of (1.1) is reduced to the discussion of a plane, non–autonomous system by considering thesystem restricted to the attractive invariant manifold.

However there are a few points to be made when entering this line of attack. The majority of the resultson the existence of attractive invariant manifolds are based on the discussion of Lyapunov type numbersof solutions, hence set in a more abstract framework4 rather than an applicable form. For the purposeof this work an approach where assumptions are made on known quantities (as the vector field) is moreconvenient.

The general setting for this case can be found in a result by Kirchgraber [9]. It supplies the existenceand additional properties of an attractive invariant manifold for mappings without giving smoothness,however. In this work we will apply an adaption by Nipp / Stoffer [13] which deals with ODE’s andestablishes smoothness as well. The assumptions on the system made by Nipp / Stoffer are expressedusing certain Lipschitz numbers of the vector field and logarithmic norms of derivatives of the vectorfield.

However, we must take into account that theLipschitz numbers of the vector field as well as the logarithmicnorms of the derivatives depend on the choice of coordinates. Hence it is of great interest to findappropriate coordinates in order to obtain satisfactory results.

Thus the difficulties in discussing the assumptions on the Lyapunov type numbers necessary for the more”abstract approach” are replaced by the problem of defining suitable coordinates, when aiming at thesetup made in [9] and [13]. The following example illustrates how the choice of ”unnatural” coordinatesmay restrict the results obtained in an unsatisfactory way.

1.3.1 Example 1 (disadvantegous cartesian coordinates)

Consider the (unperturbed) system (1.16) in the case of H(Q, P ) = P 2/2− cos(Q) of the mathematicalpendulum,

˙Q = P

˙P = − sin(Q)

H = AH,

(1.55)

where A < 0. One of the assumptions made in Nipp / Stoffer [13] includes the existence of constantsγ1 ∈ R, γ2 > 0 such that

µ(

−JD2H(Q, P ))

≤ γ1, µ (A) ≤ −γ2, γ1 < γ2, (1.56)

uniformly, where µ (M) denotes the logarithmic norm of a matrix M (cf. definition 1.4.5). Choosing theeuclidean norm on R2 one has

µ(

−JD2H(Q, P ))

= 12

(

1− cos(Q))

µ (A) = A,

4as, for instance, given in [4], [6]

1.3. Some Illustrative Examples 21

such that if 1− cos(Q) ≥ 2 |A|, then (1.56) is not satisfied. Thus the existence of an attractive manifoldmay not be established but on a subset of

(Q, P ) ∈ R2∣

∣ 1− cos(Q) < 2 |A|

depending on A. Since thehyperplane H = 0 is a global attractive invariant manifold one expects a result independent of the size ofA. Hence the cartesian coordinates (Q, P ) are ”unnatural” even in the unperturbed case ε = 0. We willsee that using certain action angle coordinates, the domain on which an attractive invariant manifoldmay be established is equal to the entire region covered by the action angle coordinates, independent ofA.

The next example illustrates a further reason of more practical nature to introduce action angle coordi-nates.

1.3.2 Example 2 (further reasons to introduce action angle coordinates)

Let us assume for a moment, that the existence of an invariant manifold Mε has been established on asufficiently large domain for a perturbed (autonomous) system of the form

˙Q = P +O(ε)

˙P = − sin(Q) +O(ε)

H = AH+O(ε),

(1.57)

(where A < 0 again). As we are interested in an explicit representation of the vector field restricted tothe manifold Mε, it will be necessary to calculate Mε. A possible line of attack consist in writing theso–called equation of invariance: Assuming that Mε is a graph of a map S, i.e.

Mε =

(Q, P ,H) ∈ R3∣

∣H = S(Q, P , ε)

we find on one hand H = ∂QS(Q, P , ε) P − ∂PS(Q, P , ε) sin(Q) +O(ε), while on the other hand (1.57)

implies H = AS(Q, P , ε) +O(ε). In general this yields a partial differential equation impossible to solvefor S explicitely, even if S is expanded with respect to ε.

Considering any region of the (Q, P )–space excluding the separatrices and fixed points of the unperturbedsystem one may define appropriate action angle coordinates such that equation (1.57) transforms into asystem of the form

ϕ = ω(h) +O(ε)

h = O(ε)

H = AH+O(ε).

(1.58)

The equation of invariance then reads

∂ϕS(ϕ, h, ε)ω(h) +O(ε) = AS(ϕ, h, ε) +O(ε).

Proving that the map S is periodic with respect to ϕ and considering its Fourierseries, an ε–expansionof S may be found explicitely by comparing Fourier coefficients in the equation of invariance. If theperturbation in (1.57) is non–autonomous, then one may proceed in a similar way, using the Ansatz

S(t, ϕ, h, ε) =∑

k,n∈Z

Sk,n(h, ε) ei(kϕ+nt).


We conclude that it is advantageous to use action angle coordinates, if possible. First since the domainwhere the existence of an attractive manifold may be established is expected to be maximal in a certainsense, second because an expansion of the invariant manifold may be found explicitely.

The third example will illustrate briefly how suitable action angle coordinates are defined. Moreover, itshows that in the case of 0 ∈ J (cf. 1.97 c and the corresponding paragraph in section 1.1.3), one has toproceed carefully if extending the domain into the periodic solution near the origin.

1.3.3 Example 3 (extending the domain of action angle coordinates)

Let H(Q, P ) = 12 (P

2 + Q2), A = −1 and the perturbation be given as follows:

˙Q = P + ε P (P − εH)

˙P = −Q− 2 εH

H = −H.

(1.59)

For p0 ∈ J = R, the corresponding Hamiltonian system admits the periodic solutions

(q, p)(t; 0, p0) = p0

(

sin(t)cos(t)

)

(1.60)

with frequency Ω(p0) = 1. Using the explicit form (1.60) we introduce action angle coordinates by setting

(Q, P ) = Φ(ϕ, h) =: P(h)

(

sin(ϕ)cos(ϕ)

)

ϕ ∈ R, h ∈ J (1.61)

where the map P is chosen appropriate and satisfies P(0) = 0. The formal transformation of system(1.59) into these new coordinates yields

ϕ = 1 + ε cos2(ϕ) (P(h) cos(ϕ)− εH) + 2 εH

P(h)sin(ϕ)

h = ε P(h)ddhP(h)

sin(ϕ) cos(ϕ) (P(h) cos(ϕ) − εH)− 2 εH

P(h)cos(ϕ)

H = −H.

(1.62)

As we are in the situation where 0 ∈ J holds, we have P(0) = 0 such that (1.62) is singular in h = 0.(Note that by definition (1.61), h = 0 corresponds to (Q, P ) = (0, 0) and therefore the periodic solution(q(t, ε), p(t, ε)) arising near the elliptic fixed point of the unperturbed system).

The extension of the action angle coordinates into h = 0 therefore may not be performed straightforward,but requires some preliminary preparations.

More precisely one may see that (1.62) is singular due to the fact that the right hand side of the (Q, P )–subsystem in (1.59) does not vanish for (Q, P ) = 0. We therefore prepare (1.59) by applying a suitabletransformation:

As the set

(Q, P ) ∈ R2∣

∣ (Q, P ) = (−εH, εH)

(1.63)

1.3. Some Illustrative Examples 23

is invariant with respect to (1.59), the transformation

(Q, P ) = (−εH, εH) + (Q,P ) (1.64)

may be performed, yielding the system

Q = P + εP (P + εH)

P = −QH = −H.

(1.65)

Here the right hand side of the (Q, P )–equation vanishes for (Q,P ) = (0, 0), hence the H–axis is invariantwith respect to (1.65). Applying (1.61) on (1.65) then yields

ϕ = 1 + ε cos2(ϕ) (P(h) cos(ϕ) + εH)

h = ε P(h)ddhP(h)

sin(ϕ) cos(ϕ) (P(h) cos(ϕ) + εH)

H = −H.

(1.66)

Following the properties of P assumed in 1.97 a, this system admits a C r+5–extension into h = 0.

1.3.4 Example 4 (reasons to introduce the map P)

Let us rewrite transformation (1.61) of example 3 for P(h) =√2 h :

(Q, P ) =√2 h

(

sin(ϕ)cos(ϕ)

)

The solution (q, p)(t; 0,√2 h) with initial value (0,

√2 h) at time t = 0 of the corresponding Hamiltonian

system satisfies H((q, p)(t; 0,√2 h)) = h for all t ∈ R. Hence for this choice of P , the action variable h

may be viewed as the ”energy” of the solutions. Although these action angle coordinates appear to besuitable, the corresponding system is not differentiable in h = 0 :

ϕ = 1 + ε cos2(ϕ)(√

2 h cos(ϕ) + εH)

h = ε h sin(ϕ) cos(ϕ)(√

2 h cos(ϕ) + εH)

H = −H.

In order to extend the corresponding system in action angle coordinates into h = 0 in a sufficient regularway, assumption 1.97 b on the map P therefore is essential.

Additionally we will see in what follows, that the region of the phase space on which the result given in[13] may be applied to, must be invariant. Due to this assumption, it may be necessary to introduce a”cutting function” in order to change the vector field locally, if dealing with regions having non-invariantboundaries. This may be achieved by choosing P in a suitable way. However, inside the regions (onany compact subset) P may be taken as the identity P(h) = h. In the case 0 ∈ J we will see that theset h = 0 is invariant with respect to the corresponding system. Hence in this situation P(h) = h isadmissible even for small h ≥ 0.


1.4 The Strongly Stable Manifold of the Equilibrium Point

Consider system (1.16) for ε = 0. In this unperturbed case the (Q, P )–hyperplane H = 0 is a centermanifold of the fixed point (Q, P ,H) = 0 (cf. figure 1.1 where d = 1).

Similarly we find the H–subspace (Q, P ) = 0 to be an

Q

P

H

Figure 1.1: The center manifold and thestable manifold in the unperturbed case

invariant manifold of (1.16). It contains all solutions lim-iting in the origin. Hence the H–space corresponds to thestable manifold of the origin. More generally it may be con-sidered as an invariant manifold which contains the originand may be represented as the graph of the constant mapRd ∋ H 7→ 0 ∈ R2.

The aim of this section is to show that in the perturbed casewhere ε 6= 0 (but small) such an invariant graph containingthe origin exists as well. More precisely we will prove theexistence of an invariant manifold of the perturbed system

which contains the origin and may be written as the graph of a (time–dependent) function

V : R× Rd × R ∋ (t,H, ε) 7→ V(t,H, ε) ∈ R2

where V(t,H, 0) = 0. As demonstrated in section 1.3.3 such an invariant manifold may be used to preparethe extension of the domain of action angle coordinates if considering regions close to an elliptic fixedpoint (i.e. the case 0 ∈ J considered in 1.97 c).

Although the definition of the stable manifold of the origin is unique in the unperturbed situation, thenotion of a stable manifold in the perturbed case may be generalized in different ways. There are basicallytwo approaches found in literature, based on different aspects of the unperturbed stable manifold:

• As the unperturbed stable manifold consist of all solutions limiting to the fixed point, the perturbedstable manifold may equally be defined as the set of all orbits appoaching the origin as t → ∞.However, since the origin is not hyperbolic in our situation various bifurcation scenarios are possibleif ε 6= 0. As for instance the origin may become globally attractive such that the stable manifold ofthe perturbed system would be given by the entire phase space.

• On the other hand, the spectrum of the linearization of the perturbed system may always be dividedinto a subset of eigenvalues with real parts of size O(ε) (i.e. the perturbed ”center”– eigenvalues)and a part of eigenvalues with real parts of size O(1) (the perturbed ”stable”– eigenvalues). Fromthis point of view, the stable manifold could be defined via the eigenspace corresponding to theperturbed ”stable”– eigenvalues. This would yield the invariant manifold which consists of thesolutions with the strongest rates of attraction towards the origin.

The definitions for the stable manifold of the perturbed system found in literature are usually based oneither of these two approaches. For our purpose it will be sufficient to content ourselves to establish theexistence of an invariant graph of a map V . Since this approach corresponds to the second approachlisted above, we will refer to this manifold as to the strongly stable manifold.

1.4. The Strongly Stable Manifold of the Equilibrium Point 25

1.4.1 The Existence of the Strongly Stable Manifold

In this first subsection we will state the existence of the strongly stable manifold of system (1.16) forsmall parameters ε. The theory found in various contributions (see [8], [10]), which may be applied toestablish the existence of a strongly stable manifold deals with the special case where the linearization ofthe perturbation vanishes at the origin.

Thus we are not in the position to apply these results directly5. However it is possible to modify theprogram carried out in [8] in a way such that the statements needed here may be established. We thereforewill not verify all details but confine ourselves with a sketch of the adapted proof strategy.

The main idea to proceed in the more general case where the linearization of the perturbation does notvanish at the origin consist in writing the map V using a linear map Vλ in the form

V(t,H, ε) := λVλ(t, ε/λ2,H)H (1.67)

where the existence of Vλ is obtained by a contraction mapping argument and λ is a sufficiently small,fixed parameter. This will be demonstrated in the proof of the following proposition :

Proposition 1.4.1 Given any > 0 there exists an ε2 = ε2(r, ) as well as a map V defined for t ∈ R,|H| < , |ε| ≤ ε2 with values in R2 and of class Cr+7 (where all derivatives up to order r+7 are uniformlybounded by 1) such that the graph

Nε :=

(t, (Q, P ),H) ∈ R× R2 × Rd∣

∣ (Q, P ) = V(t,H, ε), |H| <

(1.68)

is an invariant set of (1.16). Moreover the map V satisfies the following properties :

1. V(t, 0, ε) = 0

2. V(t,H, 0) = 0

3. V is 2π–periodic with respect to t.

The proof of this proposition is carried out in several steps.

• The first step consist in simplifying the notation as follows : Given any fixed 0 < λ < 1 we set

(x, y) := (H, (Q, P ))

ϑ := (t, ǫ) := (t, ε/λ2).(1.69)

Using these abbreviations we will rewrite system (1.16) in autonomous form. The independentvariable will be denoted by s and differentiation with respect to s is marked by a dot again (i.e. ϑ).

• Lemma 1.4.2 There exist maps X0, Y0, Y1 and Y2 defined for t ∈ R, |ǫ| < ε1, |x| < and y ∈ R2

as well as a matrix B ∈ R2×2 such that (1.16) is equivalent to the (autonomous) system

ϑ = a

x = Ax+ λ2X0(ϑ, y;λ) y

y = B y + λ2 Y0(ϑ, y;λ)x + λ2 Y1(ϑ, y;λ) y + Y2(y)(y, y)

(1.70)

5The author of this thesis did not find a way to reproduce an estimate analogous to equation (32) in [8] for the situationdiscussed there in section 6, i.e. the perturbed case. (For an illustrative example, consider the system x = −x+ε y, y = ε x.)This eventually gave rise to the modification introduced here.


for |x| < , where a =

(

10

)

. Moreover the following statements are true :

1.71 a. X0, Y0 and Y1 vanish for ϑ = (t, 0), i.e. ǫ = 0.

1.71 b. X0, Y0 and Y1 are ~ω := (2π, 0)–periodic with respect to ϑ.

1.71 c. X0, Y0 and Y1 are of class Cω. Hence there exists a b0 < ∞ such that all derivatives up toorder r + 4 are bounded by b0, uniformly with respect to t ∈ R, |x| < , y ∈ R2 and ǫ < ε1.

1.71 d. ℜ(σ(B)) = 0.

Recall that by A we denote the diagonalizable matrix of system (1.1), satisfying ℜ(σ(A)) ≤ −c0(cf. GA 1.2).

PROOF: For x = H, y = (Q, P ), ϑ = (t, ε/λ2) we define the quantities X0, Y0, Y1, Y2 and B asfollows:

X0(ϑ, y;λ) :=1

λ2

1∫

0

∂(Q,P )G(σ y, t, ǫ λ2) dσ Y0(ϑ, y;λ) :=

1

λ2∂HF (y, 0, t, ǫ λ

2)

Y1(ϑ, y;λ) :=1

λ2

1∫

0

∂(Q,P )F (σ y, 0, t, ǫ λ2) dσ Y2(y) :=

1∫

0

(1− σ)JD3H(σ y) dσ

B := JD2H(0, 0).

As shown in proposition 1.2.4 the map F vanishes for (x, y) = (0, 0) and is affine with respect tox = H. Hence taking into account that ∂HF does not depend on x we have

F (y, x, t, ε) = F (y, x, t, ε)− F (y, 0, t, ε) + F (y, 0, t, ε)− F (0, 0, t, ε)

=

1∫

0

ddσ

(

F (y, σ x, t, ε))

dσ +

1∫

0

ddσ

(

F (σ y, t, 0, ε))

dσ

=

1∫

0

∂HF (y, 0, t, ǫ λ2)x dσ +

1∫

0

∂(Q,P )F (σ y, 0, t, ǫ λ2) y dσ

= λ2Y0(ϑ, y;λ)x+ λ2Y1(ϑ, y;λ) y.

Using the integral representation of the Taylor remainder term and taking into account∇H(0, 0) = 0we find

J∇H(Q, P ) = J∇H(0, 0) + JD2H(0, 0)(

Q, P)

+

1∫

0

(1− σ)JD3H(σ y)(y, y) dσ

= B y + Y2(y)(y, y).

Additionally it follows from G(0, 0, t, ε) = 0 that

G(y, t, ε) =

1∫

0

ddσ

(

G(σ y, t, ε))

dσ =

1∫

0

∂(Q,P)G(σ y, t, ε) y dσ = λ2X0(ϑ, y;λ) y.


• In a next step we define an appropriate space for the maps V used in the ansatz (1.67) :

Definition 1.4.3 Let X j denote the following subspace of C j–maps taking values in the spaceL(Rd,R2) of d× 2–matrices :

X j :=

V ∈ C j(R× (−ε1, ε1)× Rd, L(Rd,R2))∣

∣V satisfies (1.73 a)–(1.73 c)

, (1.72)

where

1.73 a. V is ~ω–periodic with respect to ϑ

1.73 b. V (ϑ, x) = 0 if ϑ = (t, 0)

1.73 c. ‖V ‖X j <∞ with ‖V ‖X j := maxα∈N

2+d

0≤|α|≤j

supt∈R

|ǫ|≤ε1

sup|x|<

∣

∣

∣∂ α(ϑ,x)V (ϑ, x)∣

∣

∣ .

Note that for any multi–index α ∈ N 2+d, |α| := α1+ · · ·+α2+d and ∂ α(ϑ,x) := ∂ α1t ∂ α2

ǫ ∂ α3x1

. . . ∂α2+dxd .

Then (X j , ‖.‖X j) is a Banach space.

• For any V ∈ X r+7 we substitute y = λV (ϑ, x)x into the perturbation terms of (1.70), i.e. considerthe systems

ϑ = a

x = Ax+ λ3X0(ϑ, λ V (ϑ, x)x;λ)V (ϑ, x)x(1.74)

and

y = B y + λ2 Y0(ϑ, λ V (ϑ, x)x;λ)x + λ3 Y1(ϑ, λ V (ϑ, x)x;λ)V (ϑ, x)x

+ λ2Y2(λV (ϑ, x)x)(V (ϑ, x)x, V (ϑ, x)x).(1.75)

Let (ϑ, x)(s) := (ϑ, x)(s;ϑ0, x0;V ) denote the solution of (1.74) with initial value (ϑ0, x0) at times = 0 (where ϑ0 := (t0, ε0)) depending on V . We then will show that there exists a Vλ ∈ X r+7,such that

y(s) := λVλ((ϑ, x)(s;ϑ0, x0;Vλ)))x(s)

is a solution of (1.75) for V = Vλ. This, however implies immediately that (ϑ, x, y)(s) is a solutionof (1.70). We will establish the existence of such a Vλ in an analogous way to the process given in[8]. In particular the rescalation parameter λ is necessary to obtain sufficient regularity.

• For any fixed V ∈ BX r+8(1) where

BX r+8(1) :=

V ∈ X r+8∣

∣ ‖V ‖X r+8 ≤ 1

the following lemma presents a result on the fundamental solutions associated with (1.74):

Lemma 1.4.4 For any initial value (ϑ0, x0) and any V ∈ BX r+8(1) let U(s) = U(s;ϑ0, x0;V )denote the unique solution of

U(s) =(

A+ λ3X0(ϑ(s), λ V (ϑ(s), x(s))x(s);λ) V (ϑ(s), x(s)))

U(s) (1.76)

satisfying U(0) = IRd . Then x(s;ϑ0, x0;V ) = U(s;ϑ0, x0;V )x0.

Moreover there exists λ1 > 0 and a polynomial π(s) with positive coefficients such that for 0 < λ <λ1 and |x0| < ,

|U(s;ϑ0, x0;V )| ≤ e−c02 s

∣

∣

∣∂ α(ϑ0,x0)U(s;ϑ0, x0;V )

∣

∣

∣ ≤ e−c02 s λ3π(s) 0 < |α| ≤ r + 8.


This lemma 1.4.4 is proved by induction with respect to the length |α| of the multi–index α. Theinduction is carried out using the notion of the logarithmic norm (introduced in the followingdefinition 1.4.5) and the statement given in lemma 1.4.6 :

Definition 1.4.5 Following Stroem [18] we introduce the so–called logarithmic norm of a matrixM ∈ Rn×n by

µ (M) := limδ→0+

|IRn + δM | − 1

δ,

where |.| denotes the matrix norm based on the norm chosen on Rn.

As a simple consequence of lemma 2 in [18] we find

Lemma 1.4.6 Consider a solution W (s) of the inhomogenous, non–autonomous linear differentialequation

W (s) =M(s)W (s) +N(s)

where M(s), N(s) are time–dependent linear operators on Rd, the logarithmic norm µ(M(s)) is

uniformly bounded by − c02 and |N(s)| ≤ λ3 e−

c02 sπ(s) (π is a polynomial with positive coefficients).

Then

|W (s)| ≤ e−c02 s(

|W (0)|+ λ3 π(s))

s ≥ 0.

where π(s) =s∫

0

π(t) dt has positive coefficients as well.

• As mentioned above, the existence of a map Vλ defining an invariant manifold (see (1.67), (1.68))is established using the contraction mapping theorem. The definition of the mapping consideredand the proof of its contracting properties are the subject of the next step in this line:

Lemma 1.4.7 There exists a λ2 := λ2(r, ) > 0 such that for every V ∈ BX r+8(1), 0 < λ < λ2,the image TV of the map T , given by

TV (ϑ0, x0) =− 1λ

∫ ∞

0

e−sB(

λ2 Y0(ϑ, λ V (ϑ, x)x;λ)U

+ λ3 Y1(ϑ, λ V (ϑ, x)x;λ)V (ϑ, x)U

+ λ2Y2(λV (ϑ, x)x)(V (ϑ, x)x, V (ϑ, x)U))

ds

(1.77)

exists. Recall that (ϑ, x)(s) = (ϑ, x)(s;ϑ0, x0;V ), U(s) = U(s;ϑ0, x0;V ) denote solutions of (1.74),(1.76) respectively.

Moreover, the map T is a contraction from BX r+8(1) to BX r+8(1) with respect to the X r+7–topologyinduced on X r+8, i.e.

1.78 a. TV ∈ BX r+8(1)

1.78 b. ‖TV1 − TV2‖X r+7 ≤ 12 ‖V1 − V2‖X r+7 for all V1, V2 in BX r+8(1).

The way followed to establish this statement is similar to the one given in [8], p. 558–561. Theestimates found in lemma 1.4.4 are used repeatedly. Furthermore one has to apply lemma 1.4.6 toderive the scalar bounds for ∂ α(ϑ,x)TV , ∂ α(ϑ,x) (TV1 − TV2), respectively.


• In order to complete the proof of proposition 1.4.1, let Vλ ∈ X r+7 denote the unique fixed pointof T , which exists by the contraction mapping theorem. Then the group property of the flow(ϑ, x)(s; . , . ;Vλ), i.e.

(ϑ, x)(s; (ϑ, x)(s;ϑ0, x0;Vλ);Vλ) = (ϑ, x)(s + s;ϑ0, x0;Vλ)

together with Vλ = TVλ implies that the function

y(s;ϑ0, x0;Vλ) := λVλ((ϑ, x)(s;ϑ0, x0;Vλ))x(s;ϑ0, x0;Vλ)

satisfies(1.75). Hence it eventually follows that fixing any 0 < λ < λ2 and setting ε2 := ε1 λ2, the

map

V(t,H, ε) := λVλ(t, ε/λ2,H)H t ∈ R, |H| < , |ε| < ε2 (1.79)

defines an invariant manifold with the properties claimed in proposition 1.4.1.

The following remark on the parametrization V of the strongly stable manifold will help us to find anappropriate representation of the vector field when performing a transformation into the strongly stablemanifold (see next section).

Remark 1.4.8 The map V asserted in proposition 1.4.1 satisfies the following partial differential equation

∂tV(t,H, ε) = J∇H(V(t,H, ε)) + F (V(t,H, ε),H, t, ε)− ∂HV(t,H, ε)(

AH+ G(V(t,H, ε), t, ε))

.

PROOF: Since for any solution (Q, P ) = V(t,H, ε) of (1.16) we have

( ˙Q, ˙P ) = J∇H(V(t,H, ε)) + F (V(t,H, ε),H, t, ε)

=d

dtV(t,H, ε)

= ∂tV(t,H, ε) + ∂HV(t,H, ε)(

AH+ G(V(t,H, ε), t, ε))

independent of the solution (Q, P ) considered, the statement follows at once.


1.4.2 The Transformation into the Strongly Stable Manifold

The aim of this section is to transform the ”H–axis” (Q, P ) = 0 of system (1.16) ”into the stronglystable manifold” Nε (as motivated in (1.64)). We will denote the new coordinates by (Q,P ) and calculatethe transformed vector field of (1.16) with respect to these new coordinates. As seen in section 1.3.3we then expect the H–axis (Q,P ) = (0, 0) to be invariant with respect to the transformed system. Inorder to prepare the discussions to follow, we are interested in deriving representations of the transformedvector field, similar to (1.17). Hence we will compute the terms of order O(ε) and O(ε2) in an explicitform.

The leading ε–terms of V may be calculated in an easy way using the contraction T introduced in (1.77).More precisely one has to expand the fixed point equation Vλ(t0, ǫ0, x0) = TVλ(t0, ǫ0, x0) with respect toǫ0. Taking into account that D3H(0, 0) = 0 (GA 1.1a) one then applies (1.69), (1.79) to (1.77), yieldingthe identity

V(t0,H0, ε) = εV1(t0)H0 + ε2 V2(t0,H0)H0 + ε3 V3(t0,H0, ε)H0 (1.80)

where

V1(t0) =

∫ 0

∞e−sB ∂HF

1(0, 0, 0, s+ t0) esA ds

V2(t0,H0) =

∫ 0

∞e−sB

(

∂HF2(0, 0, 0, s+ t0) + ∂(Q,P )F

1(0, 0, 0, s+ t0)V1(s+ t0))

esA

+ e−sB ∂H∂(Q,P )F1(0, 0, 0, s+ t0)

(

V1(s+ t0)H0, esA)

ds.

(1.81)

As assumed in GA 1.1a, GA 1.2 the matrices A and B are diagonalizable such that the exponentials esA,e−sB admit the representation

esA =∑

λ∈σ(A)

es λ TA,λ TA,λ ∈ C d×d

e−sB =∑

ω∈σ(B)

e−sω TB,ω TB,ω ∈ C 2×2(1.82)

and the eigenvalues λ ∈ σ(A) have all negative real part, the eigenvalues ω ∈ σ(B) purely imaginary. Ina straightforward calculation one therefore obtains from (1.20)

V1(t0) =∑

|n|≤Neint0 V1

n

V2(t0,H0) = V20 (t0) + V2

1 (t0,H0) :=∑

|n|≤2N

(

V2n,0 + V2

n,1(H0))

eint0(1.83)

where V21 is linear with respect to H0 and we have set

V1n :=

∑

λ∈σ(A)ω∈σ(B)

(in− ω + λ)−1 TB,ω ∂HF1n(0, 0, 0)TA,λ (1.84)


and

V2n,0 :=

∑


(in− ω + λ)−1 TB,ω ∂HF2n(0, 0, 0)TA,λ

+∑

|n|,|n|≤Nn+n=n

∑


(i(n+ n)− ω + λ)−1 TB,ω ∂(Q,P )F1n(0, 0, 0)V1

n TA,λ

V2n,1(H0) :=

∑

|n|,|n|≤Nn+n=n

∑


(i(n+ n)− ω + λ)−1

TB,ω ∂H∂(Q,P )F1n(0, 0, 0)(V1

nH0, TA,λ).

(1.85)

We now are in the position to introduce the transformation announced and to derive an explicit formulafor the ε–expansion of the transformed vector field.

Proposition 1.4.9 For any > 0, t ∈ R, |H| < and ε < ε2(r, ) we consider the change of coordinatesgiven by

((Q, P ),H, t, ε) = ((Q,P ) + V(t,H, ε),H, t, ε). (1.86)

Then the following statements are true:

• System (1.16) transforms into

(Q, P ) = J∇H(Q,P ) + F (Q,P,H, t, ε)

H = AH+ G(Q,P,H, t, ε),(1.87)

where the maps F , G are of class Cr+7, 2π–periodic with respect to t and

F (0, 0,H, t, ε) = 0 G(0, 0, 0, t, ε) = 0

F (Q,P,H, t, 0) = 0 G(Q,P,H, t, 0) = 0.(1.88)

• The mappings F , G admit a representation6 of the form

F (Q,P,H, t, ε) =3∑

j=1

εj F j(Q,P,H, t) + ε4 F 4(Q,P,H, t, ε)

G(Q,P,H, t, ε) =

2∑

j=1

εj Gj(Q,P,H, t) + ε3 G3(Q,P,H, t, ε)

(1.89)

6for the application in chapter 4 it suffices to consider the expansions including terms of order O(ε2) of F and of order

O(ε) of G.


and more explicitely

F 1(Q,P,H, t) = J(

D2H(Q,P )−D2H(0, 0))

V1(t)H

+ F 1(Q,P,H, t)− F 1(0, 0,H, t)

F 2(Q,P,H, t) = J(

D2H(Q,P )−D2H(0, 0))

V2(t,H)H

+ 12JD

3H(Q,P )(V1(t)H)[2]

+ F 2(Q,P,H, t)− F 2(0, 0,H, t)

+(

∂(Q,P )F1(Q,P,H, t)− ∂(Q,P )F

1(0, 0,H, t))

V1(t)H

− V1(t) G1(Q,P, t)

(1.90)

as well as

G1(Q,P,H, t) = G1(Q,P, t)

G2(Q,P,H, t) = G2(Q,P, t) + ∂(Q,P )G1(Q,P, t)V1(t)H.

(1.91)

• The map F 3 may be written in the form

F 3(Q,P,H, t) =F 3(Q,P, 0, t)− F 3(0, 0, 0, t)

− V1(t)(

G2(Q,P, t)− G2(0, 0, t))

+ F 3,1(Q,P,H, t)H(1.92)

for a suitable map F 3,1 : R2 × Rd × R → L(Rd,R2).

• Finally, F 1, F 2, G1 and G2 may be represented as Fourier polynomials in t, i.e.

F j(Q,P,H, t) =∑

|n|≤jNF jn(Q,P,H, t) e

int Gj(Q,P,H, t) =∑

|n|≤jNGjn(Q,P,H, t) e

int.

(1.93)

Note that although we write H in the arguments of G1 in (1.89) for simplicity, this map does not dependon H.

PROOF: Taking the time derivative of transformation (1.86) and using (1.16) we find

(Q, P ) = J∇H((Q,P ) + V(t,H, ε)) + F ((Q,P ) + V(t,H, ε),H, t, ε)−∂tV(t,H, ε)− ∂HV(t,H, ε)

[

AH+ G((Q,P ) + V(t,H, ε), t, ε)]

.

which together with the identity found for ∂tV(t,H, ε) in remark 1.4.8 yields

(Q, P ) = J∇H((Q,P ) + V(t,H, ε))− J∇H(V(t,H, ε))+F ((Q,P ) + V(t,H, ε),H, t, ε)− F (V(t,H, ε),H, t, ε)−∂HV(t,H, ε)

[

G((Q,P ) + V(t,H, ε), t, ε)− G(V(t,H, ε), t, ε)]

.

Setting

F (Q,P,H, t, ε) := J∇H((Q,P ) + V(t,H, ε))− J∇H(V(t,H, ε))− J∇H(Q,P )

+ F ((Q,P ) + V(t,H, ε),H, t, ε)− F (V(t,H, ε),H, t, ε)− ∂HV(t,H, ε)

[

G((Q,P ) + V(t,H, ε), t, ε)− G(V(t,H, ε), t, ε)]

(1.94)


we find F to be of class Cr+7 (since V ∈ Cr+7) and

(Q, P ) = J∇H(Q,P ) + F (Q,P,H, t, ε).

Expanding F with respect to V(t,H, ε) yieldsF (Q,P,H, t, ε) =

(

JD2H(Q,P )− JD2H(0, 0))

V(t,H, ε)+ 1

2

(

JD3H(Q,P )− JD3H(0, 0))

V(t,H, ε)[2]

+O(V(t,H, ε)[3])

+F (Q,P,H, t, ε)− F (0, 0,H, t, ε)

+(

∂(Q,P )F (Q,P,H, t, ε)− ∂(Q,P )F (0, 0,H, t, ε))

V(t,H, ε)

+ 12

(

∂2(Q,P )

F (Q,P,H, t, ε)− ∂2(Q,P )

F (0, 0,H, t, ε))

V(t,H, ε)[2]

+O(V(t,H, ε)[3])

−∂HV(t,H, ε)[

G(Q,P, t, ε)− G(0, 0, t, ε)

+(

∂(Q,P )G(Q,P, t, ε)− ∂(Q,P )G(0, 0, t, ε))

V(t,H, ε)

+ 12

(

∂2(Q,P )

G(Q,P, t, ε)− ∂2(Q,P )

G(0, 0, t, ε))

V(t,H, ε)[2]

+O(V(t,H, ε)[3])]

.

Plugging in the expansion of V(t,H, ε) as given in (1.80), i.e.

V(t,H, ε) = εV1(t)H+ ε2 V2(t,H)H+ ε3 V3(t,H, ε),

we conclude

F (Q,P,H, t, ε) = ε[ (

JD2H(Q,P )− JD2H(0, 0))

V1(t)H

+(

F 1(Q,P,H, t)− F 1(0, 0,H, t)) ]

+ε2[ (

JD2H(Q,P )− JD2H(0, 0))

V2(t,H)H

+ 12

(

JD3H(Q,P )− JD3H(0, 0)) (

V1(t)H)[2]

+(

F 2(Q,P,H, t)− F 2(0, 0,H, t))

+(

∂(Q,P )F1(Q,P,H, t)− ∂(Q,P )F

1(0, 0,H, t))

V1(t)H

−V1(t)(

G1(Q,P, t)− G1(0, 0, t)) ]

+ε3[ (

F 3(Q,P, 0, t)− F 3(0, 0, 0, t)− V1(t)(

G2(Q,P, t)− G2(0, 0, t))) ]

+ε3O(H) +O(ε4).

(Take into account that the terms included in O(V(t,H, ε)[3]) are of order ε3 or higher and vanish forH = 0).

Since D3H(0, 0) = 0, F 3(0, 0, 0, t) = 0 (cf. GA 1.1b, proposition 1.2.4) the formulae (1.90), (1.92) givenin the claim are established. The representation of G(Q,P,H, t, ε) is found in an easier way :

H = AH+ G(Q, P , t, ε)

= AH+ G((Q,P ) + V(t,H, ε), t, ε).


Define G(Q,P,H, t, ε) := G((Q,P ) + V(t,H, ε), t, ε), then G ∈ Cr+7, H = AH+ G(Q,P,H, t, ε) and

G(Q,P,H, t, ε) = G(Q,P, t, ε) + ∂(Q,P)G(Q,P, t, ε)V(t,H, ε)+ 1

2∂2(Q,P )

G(Q,P, t, ε)V(t,H, ε)[2] +O(V(t,H, ε)[3])= ε G1(Q,P, t) + ε2

[

G2(Q,P, t) + ∂(Q,P )G1(Q,P, t)V1(t)H

]

+O(ε3)

which corresponds to (1.91).

The last statement of proposition 1.4.9 is obtained by plugging (1.93) and (1.83) into the representations(1.90), (1.91) respectively.

Note that since we have used the non–autonomous representation (1.16), the independent variable cor-responds to t again. Hence Q etc. denote the derivatives with respect to t.

Remark 1.4.10 It may be readily seen that if substituting F , G by F , G system (1.87) fulfills theassumptions made in GA 1.1–GA 1.3. By consequence of the transformations carried out the identities(1.88) hold and the vector fields F , G are of class Cr+7.

In the next section we will consider systems of this type in general and introduce action angle coordinates.

1.5. The Action Angle Coordinates 35

1.5 The Action Angle Coordinates

In this section we present a possible way to introduce action angle coordinates in regions of periodicsolutions of plane Hamiltonian systems. These action angle coordinates will be helpful to establish theexistence of an attractive invariant manifold and to apply averaging methods on (1.87). However thesteps carried out in this section may be applied on any system of the form

(Q, P ) = J∇H(Q,P ) + F (Q,P,H, t, ε)

H = AH+ G(Q,P,H, t, ε),(1.95)

provided that replacing F , G by F , G, the properties assumed in GA 1.1–GA 1.3 are fulfilled, F , G areof class Cr+7 and

F (0, 0,H, t, ε) = 0 G(0, 0, 0, t, ε) = 0 (1.96)

holds as well (cf. remark 1.4.10).

In the first section 1.5.1 we define the action angle coordinates and discuss some of their properties.In section 1.5.2 we then introduce a system in action angle coordinates being equivalent to (1.95) in asense. As we are interested in considering regions close to the fixed point (Q,P,H) = (0, 0, 0) as well,we eventually will show that the system introduced provides sufficient information on the qualitativebehaviour of (1.95) in a neighbourhood of the origin. The purpose of the last section 1.5.3 is to givean alternative representation of the system in action angle coordinates, aiming at the discussion of thestability of the origin. Moreover we will prove a result on the regularity of this vector field.

1.5.1 The Definition of the Action Angle Coordinates

Consider an interval J as in GA 1.1b such that the solutions (q, p)(t; 0, p0) of (1.2) with initial value(0, p0), p0 ∈ J at time t = 0 are periodic in t with frequency Ω(p0) > 0. The initial values of these periodicsolutions give rise to the definition of the action–coordinate. However we admit the action–coordinate hnot necessarily to correspond to p0 directly but to be defined via a further transformation, i.e. p0 = P(h).For instance, such a change of coordinates may consist in mapping the initial values p0 into the energyH(0, p0) of the corresponding solutions. As seen in section 1.3.4 this possibly causes regularity problems.If no transformation is performed at all (i.e. P(h) = h) then the domain of the action–coordinates dependson J . We prefer the domain of the action–coordinate h to be R, thus independent of J . As we will seein what follows, it is not necessary to fix the transformation any further at all. Therefore we considerany mapping P which fulfills the following properties:

1.97 a. P ∈ Cω(R,R)

1.97 b. P : R → J is bijective and ddhP(h) 6= 0 for h 6= 0.

1.97 c. If 0 ∈ J then P(0) = 0.

1.97 d. All the derivatives dk

dhkP(h), 1 ≤ k ≤ r + 5 are bounded uniformly with respect to h.

The angle–coordinate ϕ basically corresponds to the time variable of the periodic solutions of (1.2)considered. Although the periods Ω of these solutions generally depend on the initial value P(h), theangle coordinate ϕ is introduced in a way such that it is 2π–periodic, independent of the particularsolution.


Using the solutions (q, p)(t; q0, p0) of the Hamiltonian system (1.2) we introduce a map Φ as follows:

Definition 1.5.1 Consider the maps Ω and P as in GA 1.1b, 1.97 a. We define the following quantities:

1. For any ϕ, h ∈ R let (q, p) (ϕ, p0) := (q, p)( ϕΩ(p0)

; 0, p0) and set

Φ(ϕ, h) := (q, p)(ϕ,P(h)). (1.98)

2. In order to shorten the notation we introduce the map

ω(h) := Ω(P(h)). (1.99)

The first lemma in this section gives a summary of a few properties of the map Φ.

Lemma 1.5.2 The following statements on the maps Ω, Φ are true:

1. The map Φ is of class Cω(R2,R2) and 2π–periodic with respect to ϕ ∈ R.

2. If 0 ∈ J then

Φ(ϕ, 0) = 0. (1.100)

3. Let Ω0 denote the quantity introduced in GA 1.1a. Then

Ω(0) = Ω0. (1.101)

4. For all (ϕ, h) ∈ R2 the Jacobian determinant of Φ satisfies

detDΦ(ϕ, h) = ω(h)−1 ddhH(0,P(h)). (1.102)

For 0 ∈ J this determinant tends towards zero, i.e. detDΦ(ϕ, h) → 0 as h→ 0.

PROOF: The first two statements are simple consequences of GA 1.1 and 1.97 a together with thedefinition of Φ. We therefore content ourselves with the proof of assertions 3 and 4.

In order to establish (1.101), let us rescale the (q, p)–coordinates of system (1.2) with a parameter λ > 0:

(q, p) = (λq, λp).

We rewrite the right hand side of system (1.2) in the form of a Taylor polynomial using the integralformula for the remainder term which in addition with ∇H(0, 0) = 0 yields the expression

(

˙q˙p

)

= JD2H(0, 0)

(

qp

)

+ λ

1∫

0

(1− σ)J D3H(σλq, σλp)(q, p)[2] dσ. (1.103)

Let (q, p)(t; 0, p0, λ) denote the solution of (1.103) with initial value (0, p0) at time t = 0, where λ maytake any real value. Consider any p0 ∈ J . Then the function (q, p)(t; 0, λ p0) is a solution of (1.2), with


frequency Ω(λ p0), as it follows from GA 1.1b. Since λ (q, p)(t; 0, p0, λ) = (q, p)(t; 0, λ p0), (q, p)(t; 0, p0, λ)has frequency Ω(λ p0), too. For λ = 0 we find by (1.103)

(

q(t; 0, p0, 0)p(t; 0, p0, 0)

)

= et J D2H(0,0)

(

0p0

)

=

cos(Ω0 t)

√

∂2pH(0,0)

∂2qH(0,0) sin(Ω0 t)

−√

∂2qH(0,0)

∂2pH(0,0) sin(Ω0 t) cos(Ω0 t)

(

0p0

)

(1.104)

Here we have used the assumptions made in GA 1.1a. Thus the frequency Ω(λ p0) of (q, p)(t; 0, p0, λ)tends towards Ω0 as λ→ 0, i.e. Ω(0) = Ω0 indeed.

Let us establish the last statement claimed. By definition (1.98) of Φ we have to calculate

detDΦ(ϕ, h) = det

[

∂ϕq(ϕ,P(h)) ddh q(ϕ,P(h))

∂ϕp(ϕ,P(h)) ddh p(ϕ,P(h))

]

where

∂ϕq(ϕ,P(h)) = ω(h)−1∂tq(ϕ

ω(h) ; 0,P(h)) = ω(h)−1∂pH(Φ(ϕ, h))

∂ϕp(ϕ,P(h)) = ω(h)−1∂tp(ϕ

ω(h) ; 0,P(h)) = −ω(h)−1∂qH(Φ(ϕ, h)),(1.105)

hence

detDΦ(ϕ, h) = ω(h)−1

(

∂pH(Φ(ϕ, h)) ddh p(ϕ,P(h)) + ∂qH(Φ(ϕ, h)) d

dh q(ϕ,P(h))

)

= ω(h)−1 ddhH(Φ(ϕ, h)).

As H is the Hamiltonian of (1.2),

H(Φ(ϕ, h)) = H((q, p)( ϕω(h) ; 0,P(h))) = H((q, p)(0; 0,P(h))) = H(0,P(h)), (1.106)

thus

ddhH(Φ(ϕ, h)) = d

dhH(0,P(h)), (1.107)

proving (1.102). For 0 ∈ J we have ω(0) = Ω(0) = Ω0 6= 0 and since ∇H(0, 0) = 0

limh→0

ddhH(0,P(h)) = lim

h→0

ddh

(

O(P(h)2))

= 0.

Hence the proof of lemma 1.5.2 is complete.


By consequence of GA 1.1c the following images of Φ are well defined:

Definition 1.5.3 Let (q, p), Φ be the maps introduced in definition 1.5.1. Then we set

LJ := Φ(R,R) LJl := (q, p) (R,Jl) LJr := (q, p) (R,Jr). (1.108)

The indices J , Jl, Jr will remind us on the dependence of these quantities on the corresponding sets.

In figure 1.2 we have illustrated the situation in the case of the mathematical pendulum H(Q,P ) =

P 2/2 +(

a2

)2(1− cos(Q)) for two choices of the set J , denoted by Ju, Jc.

a

h

L

h

J

Φ(ϕ, )

Φ(ϕ, )

uc

uJ

J L

c−π π

J

P

Q

LJ

−π π ϕ

h

= LJ

−π π ϕ

h

L J

L J

r

l

l r

Figure 1.2: Illustration of the map Φ(ϕ, h) in the case of the mathematical pendulum

For the first choice Ju the domain of Φ depicted on the left hand side is mapped into a subset composedby orbits of rotatory solutions of the pendulum equation, above the separatrix. The lower and upperboundaries of the range LJ , i.e. LJl and LJr are distinct. Moreover we see that the two hatchedsubregions of the domain are mapped into two different ”strips” contained in LJ .

In the second case Jc we consider a map P satisfying P(h) = −P(−h) such that the images of the twoshaded subregions of the domain coincide. Due to the same reason, the sets LJl and LJr are identical.Moreover we emphasize that the origin (Q,P ) = (0, 0) is contained in the range LJ .


Remark 1.5.4 The study of the last assertion in lemma 1.5.2 reveals the following remarkable facts:

1. As one might be interested in choosing the map P in a way such that the transformation Φ iscanonical, we consider the case detDΦ(ϕ, h) = 1. Then lemma 1.5.2 implies that P must be chosenin a way such that

H(0,P(h)) =

h∫

0

ω(σ) dσ +H(0,P(0)) (1.109)

holds for all h ∈ R.

2. Considering the situation 0 ∈ J (hence P(0) = 0) and the special case where H(Q,P ) = P 2/2 +V (Q) and V (0) = 0 we see that (1.109) is equivalent to

P(h) =

√

√

√

√

√2

h∫

0

ω(σ) dσ h ≥ 0.

Since ω(0) 6= 0 this P is not differentiable in h = 0 such that the regularity property assumedin 1.97 a is not fulfilled. Consequently it is impossible to find a canonical transformation intothese action angle coordinates, that may be extended in a C1–manner for h = 0 (cf. section 1.3.4).

Although it is in general not possible to choose P as to make the transformation Φ canonical, sufficientinformation on system (1.1) is still afforded by introducing these coordinates.

1.5.2 The Equivalent System in Action Angle Coordinates

Taking into account that the map Φ(ϕ, h) is 2π–periodic with respect to ϕ together with the result foundon detDΦ for h → 0 in lemma 1.5.2 we see that Φ does not define a proper transformation in all cases.Hence it would not be correct to set (Q,P ) = Φ(ϕ, h) and then transform system (1.87) straightforwardinto action angle coordinates, as this would require the inverse Φ−1 of Φ.

Although this direct method is not possible, there exists a convenient way to deal with action anglecoordinates for our purpose. In fact we will consider a system in (ϕ, h)–coordinates which is qualitativelyequivalent to (1.95) ((1.87) respectively). The results found during the qualitative discussion of this newsystem then may be ”mapped” into the (Q,P )–coordinates by Φ. As it will be seen, this still providessufficient information on the qualitative behaviour of (1.95).

Definition 1.5.5 Let ( .| .) denote the euclidean inner product of Rn. Then we define the maps

F2(t, ϕ, h,H, ε) := ω(h) + ω(h)ddhH(0,P(h))

(

J ∂hΦ(ϕ, h)∣

∣

∣F (Φ(ϕ, h),H, t, ε))

F3(t, ϕ, h,H, ε) :=1

ddhH(0,P(h))

(

∇H(Φ(ϕ, h))∣

∣

∣F (Φ(ϕ, h),H, t, ε))

.(1.110)

The following lemma gives a sufficiently precise statement on the qualitative equivalence of (1.95) and asystem in action angle coordinates, defined via the maps F2 and F3:


Lemma 1.5.6 Let (t(s), ϕ(s), h(s),H(s)) be any solution of the autonomous system

dds (t, ϕ, h) =

1F2(t, ϕ, h,H, ε)F3(t, ϕ, h,H, ε)

ddsH = AH+ G(Φ(ϕ, h),H, t, ε).

(1.111)

1. Define

(Q(t), P (t)) := Φ(ϕ(t), h(t)). (1.112)

Then the map t 7→ (Q(t− t(0)), P (t− t(0)),H(t− t(0))) solves (1.95) with initial value(Φ(ϕ(0), h(0)),H(0)) at time t = t(0) and (Q(t− t(0)), P (t− t(0)),H(t− t(0))) ∈ LJ × Rd for allt ∈ R.

2. If 0 ∈ J and limt→∞

h(t) = 0 then limt→∞

(Q(t), P (t)) = (0, 0).

3. If limt→∞

h(t) = ∞ then limt→∞

dist ((Q(t), P (t)) ,LJr ) = 0 and

if limt→∞

h(t) = −∞ then limt→∞

dist ((Q(t), P (t)) ,LJl) = 0.

PROOF: It follows from dds t(s) = 1 that t(s) = s+ t(0) and thus

d

dsH(t(s)− t(0)) = d

dsH(s) = AH(s) + G(Φ(ϕ(s), h(s)),H(s), t(s), ε)

= AH(t(s)− t(0)) + G(Q(t(s)− t(0)), P (t(s) − t(0)),H(t(s)− t(0)), t(s), ε)

such thatd

dtH(t− t(0)) = AH(t− t(0)) + G(Q(t− t(0)), P (t− t(0)),H(t− t(0)), t, ε).

On the other hand we have

d

ds(Q(t(s)− t(0)), P (t(s)− t(0))) =

d

dsΦ(ϕ(t(s) − t(0)), h(t(s)− t(0)))

= DΦ(ϕ(t(s)− t(0)), h(t(s) − t(0))) dds (ϕ(s), h(s))

= DΦ(ϕ(t(s)− t(0)), h(t(s) − t(0)))

(

F2(t(s), ϕ(s), h(s),H(s), ε)F3(t(s), ϕ(s), h(s),H(s), ε)

)

= DΦ(ϕ(t(s)− t(0)), h(t(s) − t(0)))

·(

F2(t(s), ϕ(t(s) − t(0)), h(t(s)− t(0)),H(t(s)− t(0)), ε)F3(t(s), ϕ(t(s) − t(0)), h(t(s)− t(0)),H(t(s)− t(0)), ε)

)

thus

d

dt(Q(t− t(0)), P (t− t(0)))

= DΦ(ϕ(t − t(0)), h(t− t(0)))

(

F2(t, ϕ(t− t(0)), h(t− t(0)),H(t− t(0)), ε)F3(t, ϕ(t− t(0)), h(t− t(0)),H(t− t(0)), ε)

)

.


Omitting the argument t− t(0) of ϕ, h, Q, P and H it suffices to show that

(

F2(t, ϕ, h,H, ε)F3(t, ϕ, h,H, ε)

)

= [DΦ(ϕ, h)]−1(

J∇H(Q,P ) + F (Q,P,H, t, ε))

, (1.113)

as this implies the first claim at once. Since DΦ(ϕ, h) ∈ R2×2 we find

[DΦ(ϕ, h)]−1

= (detDΦ(ϕ, h))−1

[

ddh p(ϕ,P(h)) − d

dh q(ϕ,P(h))−∂ϕp(ϕ,P(h)) ∂ϕq(ϕ,P(h))

]

.

Applying (1.102) and (1.105) we find

[DΦ(ϕ, h)]−1

= ω(h)ddhH(0,P(h))

[ ddh p(ϕ,P(h)) − d

dh q(ϕ,P(h))

ω(h)−1 ∂qH(Φ(ϕ, h)) ω(h)−1 ∂pH(Φ(ϕ, h))

]

= ω(h)ddhH(0,P(h))

[

(J∂hΦ(ϕ, h))T

ω(h)−1 (∇H(Φ(ϕ, h)))T

]

.

Hence for the first component of (1.113) we have to prove

F2(t, ϕ, h,H, ε) = ω(h)ddhH(0,P(h))

(

J∂hΦ(ϕ, h)| J∇H(Φ(ϕ, h)) + F (Φ(ϕ, h),H, t, ε))

= ω(h)ddhH(0,P(h))

(∂hΦ(ϕ, h)| ∇H(Φ(ϕ, h)))

+ ω(h)ddhH(0,P(h))

(

J∂hΦ(ϕ, h)| F (Φ(ϕ, h),H, t, ε))

.

Together with the identity (1.107), thus

ddhH(0,P(h)) = d

dhH(Φ(ϕ, h)) = (∂hΦ(ϕ, h)|∇H(Φ(ϕ, h)))

this turns out to be equivalent to

F2(t, ϕ, h,H, ε) = ω(h) + ω(h)ddhH(0,P(h))

(

J∂hΦ(ϕ, h)| F (Φ(ϕ, h),H, t, ε))

,

which coincides with the definition of F2 given in (1.110). In much the similar way, for the secondcomponent of (1.113) we have to establish

F3(t, ϕ, h,H, ε) = 1ddhH(0,P(h))

(∇H(Φ(ϕ, h))| J∇H(Φ(ϕ, h)))

+ 1ddhH(0,P(h))

(

∇H(Φ(ϕ, h))| F (Φ(ϕ, h),H, t, ε))

= 1ddhH(0,P(h))

(

∇H(Φ(ϕ, h))| F (Φ(ϕ, h),H, t, ε))

.

This holds again by (1.110).


Eventually, taking into account the properties of P assumed in (1.97 a)–(1.97d), together with the regu-larity of Φ, it is easy to verify the remaining two statements of the lemma.

In our study of system (1.95) we will consider initial values (Q0, P0,H0) ∈ R 2+d with (Q0, P0) ∈ LJ .In the following lemma we show how these initial values may be ”mapped back” into the action anglecoordinates:

Lemma 1.5.7 Consider a point (Q0, P0,H0) ∈ R 2+d with (Q0, P0) ∈ LJ . Then there exists ϕ0 ∈ R andh0 ∈ R such that (Φ(ϕ0, h0),H0) = (Q0, P0,H0).

If Φ(ϕ0, h0) = (Q0, P0) for any h0 6= h0 then H(0,P(h0)) = H(0,P(h0)), i.e. h0 is uniquely determinedmodulo its ”energy” H(0,P(h0)).

PROOF: Let (Q0, P0) ∈ LJ , H0 ∈ Rd be as assumed. By definition of LJ and 1.97b there exists ϕ0 ∈ R

and h0 ∈ R∗+ such that

(Q0, P0) = Φ(ϕ0, h0).

Taking into account the identity found in (1.106) we have

H(Q0, P0) = H(0,P(h0)). (1.114)

If there is h0 as assumed, then (1.106) immediately implies H(0,P(h0)) = H(0,P(h0)). This proveslemma 1.5.7.

1.5.3 On the Regularity and the Representation of the Equivalent System

In this section we give a statement on the regularity of the maps F2(t, ϕ, h,H, ε), F3(t, ϕ, h,H, ε) anddeduce an expansion with respect to P(h). The latter will be convenient for section 1.6.3 and thediscussion of the stability of the fixed point7 (Q,P,H) = (0, 0, 0). Note that the regularity may not bededuced from definition 1.5.5 directly as P(0) (and hence d

dhH(0,P(h))) may vanish. In fact P(0) = 0 ifand only if 0 ∈ J , i.e. the region LJ contains the fixed point at the origin. Let us begin by introducingthe following notation.

Definition 1.5.8 In what follows BC r(X,Y ) shall denote the vector space of all functions f : X → Ybeing r–times continuously differentiable and having k–th derivatives (k = 0, . . . , r) of finite supremum–norm.

Using this notation the main result of this subsection reads as follows:

Proposition 1.5.9 Consider any system of the form (1.95) satisfying (1.96) and GA 1.1–GA 1.3. Let be as in proposition 1.4.1 and P fulfill the properties assumed in 1.97 a–1.97 d. If in addition 0 ∈ J ,

the interval J is bounded and∂pH(0,p0)

p06= 0 for all p0 ∈ J then the following statement is true:

7Recall that if (1.95) is derived from (1.87) deduced in the preceeding sections, then the origin (Q,P,H) = (0, 0, 0)corresponds to the periodic solution of (1.1).


The mappings F2, F3, of definition 1.5.5 are of class Cr+4 for t, ϕ, h ∈ R, H ∈ BRd() and |ε| ≤ ε2 andmay be represented in the form

F2(t, ϕ, h,H, ε) = Ω0 +1ν

(

J Φ,1(ϕ)∣

∣ ∂(Q,P )F (0, 0, 0, t, ε)Φ,1(ϕ)

)

+ 1ν

(

J Φ,1(ϕ)∣

∣ ∂(Q,P )∂HF (0, 0, 0, t, ε)(H,Φ,1(ϕ))

)

+ 12P(h) 1

ν

(

J Φ,1(ϕ)∣

∣ ∂2(Q,P )F (0, 0, 0, t, ε)Φ,1(ϕ)[2]

)

+ f ,,2(ϕ,P(h),H, t, ε)H[2] + P(h) f ,,1(ϕ,P(h),H, t, ε)H+ P(h)2f ,,0(ϕ,P(h),H, t, ε) (1.115)

F3(t, ϕ, h,H, ε) =P(h)ddhP(h)

1∂2pH(0,0)

(

H ,1(ϕ)∣

∣ ∂(Q,P )F (0, 0, 0, t, ε)Φ,1(ϕ)

)

+ P(h)ddhP(h)

1∂2pH(0,0)

(

H ,1(ϕ)∣

∣ ∂(Q,P )∂HF (0, 0, 0, t, ε)(H,Φ,1(ϕ))

)

+ 12

P(h)2

ddhP(h)

1∂2pH(0,0)

(

H ,1(ϕ)∣

∣ ∂2(Q,P )F (0, 0, 0, t, ε)Φ,1(ϕ)[2]

)

+ P(h)ddhP(h)

g,,2(ϕ,P(h),H, t, ε)H[2] + P(h)2

ddhP(h)

g,,1(ϕ,P(h),H, t, ε)H+ P(h)3

ddhP(h)

g,,0(ϕ,P(h),H, t, ε) (1.116)

where f ,,j, g,,j are Cr+4, 2π–periodic with respect to t and ϕ and

ν =

√

∂2pH(0, 0)

∂2qH(0, 0)Φ,1(ϕ) =

(

ν sin(ϕ)cos(ϕ)

)

H ,1(ϕ) =

(

Ω0 sin(ϕ)∂2pH(0, 0) cos(ϕ)

)

. (1.117)

Moreover G(Φ(ϕ, h),H, t, ε) is of class Cr+4 as well, admitting a very similar representation:

G(Φ(ϕ, h),H, t, ε) = P(h) ∂(Q,P )G(0, 0, 0, t, ε)Φ,1(ϕ) + ∂HG(0, 0, 0, t, ε)H

+ h,,2(ϕ,P(h),H, t, ε)H[2] + P(h) h,,1(ϕ,P(h),H, t, ε)H+ P(h)2h,,0(ϕ,P(h),H, t, ε). (1.118)

PROOF: We begin this proof by recalling that due to GA 1.1b and lemma 1.5.2 the maps Ω and Φ areof class Cω on J , R2 respectively. However we recall that the vector fields F and G are of class Cr+7

only. In the following steps we will loose some of this differentiability as we perform expansions in orderto yield the expressions (1.115)–(1.118).

1. Using (1.101) together with GA 1.1c, the Taylor formula yields

Ω(p0) = Ω0 + p02

1∫

0

∂2p0Ω(σ p0) (1− σ) dσ

hence setting A0(p0) :=1∫

0

∂2p0Ω(σ p0) (1− σ) dσ we have

ω(h) = Ω0 + P(h)2A0(P(h)). (1.119)


2. As (Q,P ) = (0, 0) is a fixed point of the Hamiltonian system (cf. GA 1.1a) we have (q, p) (ϕ, 0) =(0, 0), thus

(q, p) (ϕ, p0) = p0 (∂p0 q, ∂p0 p)(ϕ, 0) +12 p0

2 (∂2p0 q, ∂2p0 p)(ϕ, 0)

+ 12 p0

3

1∫

0

(∂3p0 q, ∂3p0 p)(ϕ, σ p0) (1− σ)2 dσ.

(1.120)

Recall that (q, p) is 2π–periodic with respect to ϕ.

In order to find an alternative form for (∂p0 q, ∂p0 p)(ϕ, 0) we take derivatives with respect to ϕ in(1.120) obtaining

∂ϕ (q, p) (ϕ, p0) = p0 ∂ϕ(∂p0 q, ∂p0 p)(ϕ, 0) +12 p0

2 ∂ϕ(∂2p0 q, ∂

2p0 p)(ϕ, 0) +O(p30). (1.121)

On the other hand, the definition of (q, p) implies

∂ϕ (q, p) (ϕ, p0) =1

Ω(p0)∂t(q, p)(

ϕΩ(p0)

; 0, p0) =1

Ω(p0)J∇H ((q, p) (ϕ, p0))

= 1Ω(p0)

(

JD2H(0, 0) ((q, p) (ϕ, p0)) +12 JD

3H(0, 0) ((q, p) (ϕ, p0))[2]

+ 12

1∫

0

JD4H (σ (q, p) (ϕ, p0)) (1− σ)2 dσ ((q, p) (ϕ, p0))[3]

)

.

(1.122)

Plugging (1.120) into (1.122) and comparing coefficients of equal powers of p0 in (1.121) one obtains

∂ϕ(∂p0 q, ∂p0 p)(ϕ, 0) = 1Ω0JD2H(0, 0)(∂p0 q, ∂p0 p)(ϕ, 0)

∂ϕ(∂2p0 q, ∂

2p0 p)(ϕ, 0) = 1

Ω0

(

JD2H(0, 0)(∂2p0 q, ∂2p0 p)(ϕ, 0)

+JD3H(0, 0) ((∂p0 q, ∂p0 p)(ϕ, 0))[2]

)

.

Differentiating the initial condition (q, p) (0, p0) = (0, p0) with respect to p0 and evaluating thesederivatives for p0 = 0 yields the following initial conditions :

(∂p0 q, ∂p0 p)(0, 0) = (0, 1)

(∂2p0 q, ∂2p0 p)(0, 0) = (0, 0).

Taking into account D3H(0, 0) = 0 we eventually find

(∂p0 q, ∂p0 p)(ϕ, 0) = eϕΩ0

JD2H(0,0)(

01

)

(∂2p0 q, ∂2p0 p)(ϕ, 0) = 0.

The formula for etJD2H(0,0) found in (1.104), i.e.

etJD2H(0,0) =

cos(Ω0 t)

√

∂2pH(0,0)

∂2qH(0,0) sin(Ω0 t)

−√

∂2qH(0,0)

∂2pH(0,0) sin(Ω0 t) cos(Ω0 t)


implies

(∂p0 q, ∂p0 p)(ϕ, 0) =

√

∂2pH(0,0)

∂2qH(0,0) sin(ϕ)

cos(ϕ)

.

Thus setting ν :=

√

∂2pH(0,0)

∂2qH(0,0) and Φ,1(ϕ) :=

(

ν sin(ϕ)cos(ϕ)

)

together with

A1(ϕ, p0) := 12

1∫

0

(∂3p0 q, ∂3p0 p)(ϕ, σ p0) (1− σ)2 dσ

we conclude from (1.120)

Φ(ϕ, h) = (q, p)(ϕ,P(h)) = P(h)Φ,1(ϕ) + P(h)3A1(ϕ,P(h)). (1.123)

3. The identities ∇H(0, 0) = 0 and D3H(0, 0) = 0 imply together with (1.123)

∇H(Φ(ϕ, h)) = D2H(0, 0)Φ(ϕ, h) + 12

1∫

0

D4H (σΦ(ϕ, h)) (1− σ)2 dσ (Φ(ϕ, h))[3]

= P(h)D2H(0, 0)Φ,1(ϕ) + P(h)3D2H(0, 0)A1(ϕ,P(h))

+P(h)3 1

2

1∫

0

D4H (σΦ(ϕ, h)) (1− σ)2 dσ(

Φ,1(ϕ) + P(h)2A1(ϕ,P(h))

)[3]

such that when setting H ,1(ϕ) :=

(


)

,

A2(ϕ, p0) := D2H(0, 0)A1(ϕ, p0) +12

1∫

0

D4H (σ (q, p) (ϕ, p0)) (1− σ)2 dσ(

Φ,1(ϕ) + p02A1(ϕ, p0)

)[3]

we obtain

∇H(Φ(ϕ, h)) = P(h)

[

∂2qH(0, 0) 00 ∂2pH(0, 0)

] (

ν sin(ϕ)cos(ϕ)

)

+ P(h)3A2(ϕ,P(h))

= P(h)H ,1(ϕ) + P(h)3A2(ϕ,P(h)). (1.124)

The map A2 is 2π–periodic with respect to ϕ.

4. In a similar way we may write

∂pH(0,P(h)) = P(h) ∂2pH(0, 0) + 12 P(h)3

1∫

0

∂4pH(0, σP(h)) (1− σ)2 dσ

= P(h)

∂2pH(0, 0) + 12 P(h)

2

1∫

0

∂4pH(0, σP(h)) (1− σ)2 dσ

= P(h)A3(P(h)),


where A3(p0) := ∂2pH(0, 0) + 12 p0

21∫

0

∂4pH(0, σ p0) (1 − σ)2 dσ satisfies A3(0) = ∂2pH(0, 0).

Hence we conclude

ddhH(0,P(h)) = ∂pH(0,P(h)) d

dhP(h) =(

ddhP(h)

)

P(h)A3(P(h)). (1.125)

5. Since by assumption A3(P(h)) =∂pH(0,P(h))

P(h) 6= 0 the map h 7→ 1A3(P(h)) is of class Cω. This

together with 1.97 a implies that the function P(h)ddhP(h)A3(P(h))

is of class Cω for h ∈ R.

6. As F (0, 0,H, t, ε) = 0 (cf. (1.96), (1.88) respectively), we find ∂kHF (0, 0,H, t, ε) = 0 for all 0 ≤ k ≤

r + 7, hence the Taylor expansion of F (Q,P,H, t, ε) is of the form

F (Q,P,H, t, ε) = ∂(Q,P )F (0, 0, 0, t, ε) (Q,P )

+∂(Q,P )∂HF (0, 0, 0, t, ε)(H, (Q,P )) +12∂

2(Q,P )F (0, 0, 0, t, ε) (Q,P )

[2]

+A4,1((Q,P ) ,H, t, ε)(H[2], (Q,P )) +A4,2((Q,P ) ,H, t, ε)(H, (Q,P )

[2])

+A4,3((Q,P ) ,H, t, ε) (Q,P )[3]

where the maps A4,1, A4,2 and A4,3 are of class Cr+4 and 2π–periodic with respect to ϕ. Hence

we rewrite F (Φ(ϕ, h),H, t, ε) as follows:

F (Φ(ϕ, h),H, t, ε) = ∂(Q,P )F (0, 0, 0, t, ε)Φ(ϕ, h) (1.126)

+∂(Q,P )∂HF (0, 0, 0, t, ε)(H,Φ(ϕ, h)) +12∂

2(Q,P )F (0, 0, 0, t, ε)Φ(ϕ, h)

[2]

+A4,1(Φ(ϕ, h),H, t, ε)(H[2],Φ(ϕ, h)) +A4,2(Φ(ϕ, h),H, t, ε)(H,Φ(ϕ, h)

[2])

+A4,3(Φ(ϕ, h),H, t, ε)Φ(ϕ, h)[3].

7. With G(0, 0, 0, t, ε) = 0, a similar representation is found for G(Φ(ϕ, h),H, t, ε):

G(Φ(ϕ, h),H, t, ε) = ∂(Q,P )G(0, 0, 0, t, ε)Φ(ϕ, h) + ∂HG(0, 0, 0, t, ε)H (1.127)

+A5,0(Φ(ϕ, h),H, t, ε)H[2] +A5,1(Φ(ϕ, h),H, t, ε)(H,Φ(ϕ, h))

+A5,2(Φ(ϕ, h),H, t, ε)Φ(ϕ, h)[2]

where the maps A5,2, A5,1 and A5,0 are 2π–periodic with respect to ϕ.

8. Summarizing the representations (1.119), (1.123), (1.125) and (1.126) we rewrite the term presentedin (1.110) as follows

ω(h)ddhH(0,P(h))

(

J ∂hΦ(ϕ, h)∣

∣

∣F (Φ(ϕ, h), H, t, ε))

=

Ω0 + P(h)2 A0(P(h))

ddh

P(h) P(h)A3(P(h))

(

J(

ddh

P(h) Φ,1

(ϕ) + 3 ddh

P(h) P(h)2A1(ϕ,P(h)) + d

dhP(h) P(h)

3∂p0A1(ϕ,P(h))

)

∣

∣

∣

∣

∣

· · ·

· · · P(h) ∂(Q,P )F (0, 0, 0, t, ε)(

Φ,1

(ϕ) + P(h)2A1(ϕ,P(h))

)

+ P(h) ∂(Q,P )∂HF (0, 0, 0, t, ε)(

H,Φ,1

(ϕ) + P(h)2A1(ϕ,P(h))

)

+ P(h)2 1

2∂2(Q,P )F (0, 0, 0, t, ε)

(

Φ,1

(ϕ) + P(h)2A1(ϕ,P(h))

)[2]+ P(h)A4,1(Φ(ϕ, h), H, t, ε)

(

H[2],(

Φ,1

(ϕ) + P(h)2A1(ϕ,P(h))

)

)

+ P(h)2A4,2(Φ(ϕ, h), H, t, ε)

(

H,(

Φ,1

(ϕ) + P(h)2A1(ϕ,P(h))

)[2])

+ P(h)3A4,3(Φ(ϕ, h), H, t, ε)

(

Φ,1

(ϕ) + P(h)2A1(ϕ,P(h))

)[3])

=:Ω0

A3(0)

[

· · ·

· · ·

(

JΦ,1

(ϕ)

∣

∣

∣

∣

∣

∂(Q,P )F (0, 0, 0, t, ε) Φ,1

(ϕ) + ∂(Q,P )∂HF (0, 0, 0, t, ε)(

H,Φ,1

(ϕ))

+ A4,1(Φ(ϕ, h), H, t, ε)

(

H[2],Φ,1

(ϕ)

)

)

+ P(h)

(

JΦ,1

(ϕ)

∣

∣

∣

∣

∣

12∂2(Q,P )F (0, 0, 0, t, ε) Φ

,1(ϕ)

[2]+ A4,2(Φ(ϕ, h), H, t, ε)

(

H,Φ,1

(ϕ)[2])

)

+ P(h)2A6(ϕ,P(h), H, t, ε)

]


which eventually leads to the representation claimed in (1.115). As a further consequence of thisform we conclude that the map F2 is of class Cr+4 with respect to h.

9. Summarizing the representations (1.124), (1.125) and (1.126) we rewrite F3 defined in (1.110) asfollows

1ddhH(0,P(h))

(

∇H(Φ(ϕ, h))∣

∣

∣F (Φ(ϕ, h), H, t, ε))

=

1

ddh

P(h) P(h)A3(P(h))

(

P(h)H,1

(ϕ) + P(h)3A2(ϕ,P(h))

∣

∣

∣

∣

∣

· · ·

· · · P(h) ∂(Q,P )F (0, 0, 0, t, ε)(

Φ,1

(ϕ) + P(h)2A1(ϕ,P(h))

)

+ P(h) ∂(Q,P )∂HF (0, 0, 0, t, ε)(

H,Φ,1

(ϕ) + P(h)2A1(ϕ,P(h))

)

+ P(h)2 1

2∂2(Q,P )F (0, 0, 0, t, ε)

(

Φ,1

(ϕ) + P(h)2A1(ϕ,P(h))

)[2]+ P(h)A4,1(Φ(ϕ, h), H, t, ε)

(

H[2],(

Φ,1

(ϕ) + P(h)2A1(ϕ,P(h))

)

)

+ P(h)2A4,2(Φ(ϕ, h), H, t, ε)

(

H,(

Φ,1

(ϕ) + P(h)2A1(ϕ,P(h))

)[2])

+ P(h)3A4,3(Φ(ϕ, h), H, t, ε)

(

Φ,1

(ϕ) + P(h)2A1(ϕ,P(h))

)[3])

=:P(h)

ddh

P(h)A3(0)

(

H,1

(ϕ)

∣

∣

∣

∣

∣

∂(Q,P )F (0, 0, 0, t, ε) Φ,1

(ϕ) + ∂(Q,P )∂HF (0, 0, 0, t, ε)(H, Φ,1

(ϕ)) + A4,1(Φ(ϕ, h), H, t, ε)

(

H[2],Φ,1

(ϕ)

)

)

+P(h)2

ddh

P(h)A3(0)

(

H,1(ϕ)

∣

∣

∣

∣

∣

12∂2(Q,P )F (0, 0, 0, t, ε) Φ,1(ϕ)[2] + A4,2(Φ(ϕ, h), H, t, ε)

(

H,Φ,1(ϕ)[2])

)

+P(h)3

ddh

P(h)A3(0)A7(ϕ,P(h), H, t, ε).

This implies the identity claimed in (1.116). Following the same way as for F2 one finds F3 to beof class Cr+4 h as well.

The proof of (1.118) is carried out in a very similar way (using (1.127)) and therefore omitted.

It remains to establish that the maps F2, F3 are of class Cr+4 with respect to the remaining argumentst, ϕ, H and ε. This however may be deduced from the formulae in definition 1.5.5 directly.

Let us conclude this section with the following remark on the situation where 0 ∈ J :

Remark 1.5.10 Consider the case 0 ∈ J . Then (1.96) implies that the representations (1.116), (1.118)simplify to

F3(t, ϕ, 0,H, ε) = 0 G(Φ(ϕ, 0), 0, t, ε) = 0. (1.128)

In this case the set h = 0,H = 0 is therefore invariant with respect to (1.111).


1.6 The Attractive Invariant Manifold

1.6.1 The Existence of an Attractive Invariant Manifold

In this section we establish the existence of a unique attractive invariant manifold for system (1.111) byapplying the following result by Nipp / Stoffer [13]:

Lemma 1.6.1 Consider a system of the form

ddsx = f(x, y, ϑ)

ddsy = g(x, y, ϑ)

(1.129)

where the maps f and g satisfy the following set of assumptions:

1.130 a. In the domain x ∈ Rn, y ∈ Rm, ϑ ∈ E ⊂ R the map f is of class BC r (r > 0) and g has boundedderivatives of order k = 1, . . . , r. There exists y0 ∈ Rm such that the mapping (x, ϑ) 7→ g(x, y0, ϑ)is bounded uniformly. Moreover the following Lipschitz conditions hold:

|f(x, y1, ϑ1)− f(x, y2, ϑ2)| ≤ L1,2 |y1 − y2|+ L1,3 |ϑ1 − ϑ2||g(x1, y, ϑ1)− g(x2, y, ϑ2)| ≤ L2,1 |x1 − x2|+ L2,3 |ϑ1 − ϑ2| .

1.130 b. There exist constants γ1 ∈ R, γ2 > 0 such that

µ (−∂xf(x, y, ϑ)) ≤ γ1 µ (∂yg(x, y, ϑ)) ≤ −γ2,

where µ (.) denotes the logarithmic norm.

1.130 c. 2√

L1,2L2,1 < γ2 − γ1

1.130 d. L1,2L < γ2

1.130 e. (r + 1)L1,2L < γ2 − r γ1

where L :=2L2,1

γ2−γ1+√

(γ2−γ1)2−4L1,2 L2,1

.

Then for every ϑ ∈ E, system (1.129) admits a unique manifold Mϑ which may be respresented as agraph of a mapping S, i.e.

Mϑ :=

(x, y) ∈ Rn+m∣

∣ y = S(x, ϑ)

such that the following assertions hold:

1.131 a. S ∈ BC r(Rn × E,Rm) and ‖S‖∞ := sup |S(x, ϑ)| |x ∈ Rn, ϑ ∈ E < .

1.131 b. Mϑ is invariant with respect to system (1.129) in the following sense:

For every initial condition (x0, y0) ∈ Mϑ the corresponding solution satisfies

(x, y)(s;x0, y0, ϑ) ∈ Mϑ ∀s ∈ R.

1.6. The Attractive Invariant Manifold 49

1.131 c. Mϑ is attractive, i.e. there exist constants c0, c1 > 0 such that every solution of (1.129) with initialvalue (x0, y0) ∈ Rm satisfies

|y(s;x0, y0, ϑ)− S(x(s;x0 , y0, ϑ), ϑ)| ≤ c0 e−c1 s |y0 − S(x0, ϑ)|

for all s ≥ 0.

1.131 d. If f and g are ~ω ∈ Rn–periodic with respect to x, then the map S is ~ω–periodic with respect to x.

1.131 e. Moreover if there exists a subset X ∈ Rn such that X × 0 is invariant with respect to (1.129),then S(x, ϑ) = 0 for all x ∈ X, ϑ ∈ E.

Using this result we will be in the position to establish the existence of an attractive invariant manifoldfor system (1.111). We continue with the following remark on logarithmic norms :

Remark 1.6.2 Recall definition 1.4.5 of the logarithmic norm. Let us quote the following properties oflogarithmic norms, as found in [18] (Lemmata 1a, 1b and Corollary 2):

1. If the matrixM ∈ Rn×n is diagonalizable then there exists a norm on Rn such that the correspondinglogarithmic norm of M is given by the spectral abscissa, i.e.

µ (M) = maxi=1...n

ℜ(λ) |λ ∈ σ(M) .

2. Choosing the maximum norm on Rn the logarithmic norm may be expressed as follows:

µ (M) = maxi=1...n

[M ]i,i +

n∑

j=1j 6=i

∣

∣

∣[M ]i,j

∣

∣

∣

. (1.132)

3. Choosing the euclidean norm on Rn one has

µ (M) = max|x|≤1

(x|M x) . (1.133)

4. µ(M +N) ≤ µ(M) + µ(N) for M,N ∈ Rn×n.

5. µ(λM) = λµ(M) for any λ > 0, M ∈ Rn×n.

Using these explicit representations of the logarithmic norms in particular cases and taking into ac-count that (1.111) decouples if the perturbation parameter ε is zero, we prove a slight modification oflemma 1.6.1: proposition 1.6.3 states this invariant manifold result in a form which is more convenientfor application in our situation.


Proposition 1.6.3 Consider a system of the form

ddsξ = f0(ξ) + f1(ξ, y, ε)ddsy = g0(y) + g1(ξ, y, ε)

(1.134)

defined for ξ ∈ Rn, y ∈ Rm and |ε| < ε2 where the functions f0, f1, g1 are of class BC r (r > 0) and g0

has bounded derivatives of order k = 1, . . . , r. We assume that f1, g1 vanish for ε = 0 and either of thefollowing two assumptions holds:

1. There exists a permutation matrix P ∈ Rn×n such that for N :=[∥

∥∂xjf0i

∥

∥

∞]

i,j=1...nthe matrix

P−1N P is of upper triangular form and r maxi=1...nξ∈R

n

− ∂xif0i (ξ) < −max

|y|≤µ(

Dg0(y))

> 0.

2. There exists an invertible matrix P ∈ Rn×n such that the estimate

r max|ξ|≤1ξ∈R

n

−(

ξ|P−1Df0(ξ)P ξ)

< −max|y|≤

µ(

Dg0(y))

> 0

is fulfilled.

Then there exists ε3 > 0 such that for any |ε| < ε3 the result of lemma 1.6.1 applies to system (1.134)(where ε plays the role of ϑ).

PROOF: We proceed in several steps:

1. Given any constant κ > 0 and the matrix P as assumed, let ∆ denote the diagonal matrixdiag(κ, κ2, . . . , κn) ∈ Rn×n. We introduce rescaled coordinates x defined via ξ = P ∆x and setϑ := ε. It then follows that (1.134) transforms to

ddsx = ∆−1 P−1f0(P∆x) + ∆−1 P−1f1(P∆x, y, ε) =: f(x, y, ϑ)

ddsy = g0(y) + g1(P∆x, y, ε) =: g(x, y, ϑ).

(1.135)

Calculating the derivative of f(x, y, ϑ) for ϑ = 0 yields

∂xf(x, y, 0) = ∆−1 P−1Df0(P ∆x)P ∆ =[

κj

κi

[

P−1Df0(P ∆x)P]

i,j

]

i,j=1...n.

2. If the first assumption is fulfilled, we define

b0 := −max|y|≤

µ(

Dg0(y))

, b1 := maxi,j=1...n

[

P−1N P]

i,j, b2 := max

i=1...nξ∈R

n

−∂xif0i (ξ).

Note that since P is a permutation matrix b1 ≥ 0. Considering the max–norm on Rn in this case,


eq. (1.132) implies

µ (−∂xf(x, y, 0)) = maxi=1...n

ℜ [−∂xf(x, y, 0)]i,i +n∑

j=1j 6=i

∣

∣

∣[−∂xf(x, y, 0)]i,j∣

∣

∣

= maxi=1...n

−κi

κi[

P−1Df0(P ∆x)P]

i,i

+ maxi=1...n

n∑

j=1j 6=i

κj

κi

∣

∣

∣

[

P−1Df0(P ∆x)P]

i,j

∣

∣

∣

≤ maxi=1...n

−[

P−1Df0(P ∆x)P]

i,i+ maxi=1...n

n∑

j=1j 6=i

κj−i[

P−1N P]

i,j

and as P is a permutation matrix and P−1N P is assumed to be of upper triangular form,

= maxi=1...n

−[

Df0(P ∆x)]

i,i+ maxi=1...n

n∑

j=i+1

κj−i[

P−1N P]

i,j

≤ b2 + κ maxi=1...n

n∑

j=i+1

b1

≤ b2 + κn b1.

3. If the second assumption is fulfilled, one may proceed in a similar way. Define

b0 := −max|y|≤

µ(

Dg0(y))

, b1 := 0, b2 := max|ξ|≤1ξ∈R

n

−(


.

Considering the euclidean norm on Rn the representation (1.133) implies

µ (−∂xf(x, y, 0)) = max|ξ|≤1

−(

ξ|P−1Df0(P x)P ξ)

≤ max|ξ|≤1ξ∈R

n

−(


= b2.

4. Thus the inequalities

µ (−∂xf(x, y, 0)) ≤ b2 + κn b1,

µ (∂yg(x, y, 0)) = µ(

Dg0(y))

≤ max|y|≤

µ(

Dg0(y))

= −b0(1.136)

hold uniformly with respect to x ∈ Rn, y ∈ Rm and in both situations dealed with. Recall thatby assumption b0 > r b2.

5. Choose 0 < κ < min

b0r+1 ,

b0−r b22(r+1+r n b1)

and define

γ1 := max 0, b2 + κn b1+ κ γ2 := b0 − κ.

Then γ2 > 0 since 0 < κ < b0 and

µ (−∂xf(x, y, 0)) ≤ b2 + κn b1 < max 0, b2 + κn b1+ κ = γ1


as well asµ (∂yg(x, y, 0)) ≤ −b0 ≤ −b0 + κ = −γ2.

Considering the two cases

• b2 + κn b1 ≤ 0: whereγ2 − r γ1 = b0 − κ− r max 0, b2 + κn b1 − r κ = b0 − (r + 1)κ > 0

• b2 + κn b1 > 0: implyingγ2 − r γ1 = b0 − r b2 − κ (r + 1 + r n b1) >

b0−r b22 > 0,

we see that γ2 − r γ1 is always positive.

6. As the maps f and g are of class BC r where in particular r ≥ 1, it follows immediately that thereexist Lipschitz numbers L1,2, L1,3, L2,1 and L2,3 such that (1.130 a) holds uniformly with respectto x ∈ Rn, y ∈ Rm. Taking into account that f1 vanishes for ε = 0 we find L1,2 to be of size O(ε).We conclude from the previous step that setting y0 = 0 the assumptions (1.130 b) – (1.130 e) aresatisfied for ε = 0. As L1,2 depends continuously on ε there exists an ε3(γ1, γ2, r) > 0 such that forall |ε| < ε3 the assumptions (1.130b) – (1.130 e) are fulfilled as well. Therefore lemma 1.6.1 maybe applied on (1.134) proving the statement of proposition 1.6.3.

Since the perturbations in (1.111) are not bounded uniformly, we are not in the position to apply propo-sition 1.6.3 to (1.111). However if we modify the vector field for |h| > and |H| > in a way suchthat it becomes bounded, the existence of a global attractive invariant manifold Mϑ for this modifiedvector field may be proved via proposition 1.6.3. If the modified vector field is left identical to (1.111)for |h| < , |H| < this implies the existence of a set M

∣

∣

which is invariant with respect to (1.111) in

a sense. We define the modified vector field with the help of a ”cutting function” X as follows:

Given any large > 0 let χ denote a map of class BCr+4 satisfying

1.137 a. χ(s) = 1 for s <

1.137b. χ(s) = 0 for s > 2 .

With the help of this map we then define the cutting function by

X (h,H) := χ(h)χ ((H|H) /)

such that X (h,H) = 1 if |h| ≤ and |H| ≤ , and X (h,H) = 0 if |h| ≥ 2 or |H| ≥ 2 . The modifiedvector field of (1.111) then is introduced as follows:

F(t, ϕ, h,H, ε) :=

1ω(h)0

+ X (h,H)

0F2(t, ϕ, h,H, ε)− ω(h)

F3(t, ϕ, h,H, ε)

G(t, ϕ, h,H, ε) := AH+ X (h,H) G(Φ(ϕ, h),H, t, ε).

(1.138)

Note that in the case 0 ∈ J the set h = 0,H = 0 is invariant with respect to the modified vector field(1.138) (cf. the similar statement given remark 1.5.10).


Corollary 1.6.4 Given r ∈ N as in proposition 1.4.9 there exists a positive constant ε3 such that forevery |ε| < ε3 the system

dds (t, ϕ, h) = F(t, ϕ, h,H, ε)

ddsH = G(t, ϕ, h,H, ε).

(1.139)

defined for (t, ϕ, h) ∈ R3, H ∈ Rd admits a unique, attractive invariant manifold

Mε :=

(t, ϕ, h,H) ∈ R 3+d∣

∣H = S(t, ϕ, h, ε)

.

More precisely the manifold Mε fulfills the following properties:

1.140 a. S ∈ BCr+4(R3 × (−ε3, ε3),Rd).

1.140 b. Mε is invariant with respect to the system (1.139).

1.140 c. Mε is uniformly attractive.

1.140 d. The map S is 2π–periodic with respect to the variables t and ϕ.

1.140 e. For ε = 0, S(t, ϕ, h, 0) = 0 and if 0 ∈ J , then S(t, ϕ, 0, ε) = 0.

Moreover the set M∣

∣

:= Mε∩

(t, ϕ, h,H) ∈ R 3+d∣

∣ |h| < , |H| <

is invariant with respect to (1.111)

in the following sense:

Given a solution (t, ϕ, h,H) of (1.111) together with a set I ⊂ R such that for s0 ∈ I, H(s0) =S((t, ϕ, h)(s0), ε) and |h(s)| < for s ∈ I the identity

H(s) = S((t, ϕ, h)(s), ε) s ∈ I

holds.

PROOF: In order to establish the assumptions made in proposition 1.6.3 we set

ξ := (t, ϕ, h), y := H

f0(ξ) :=

1ω(h)0

f1(ξ, y, ε) := X (h,H)

0F2(t, ϕ, h,H, ε)− ω(h)

F3(t, ϕ, h,H, ε)

(1.141)

g0(y) := AH g1(ξ, y, ε) := G(t, ϕ, h,H, ε)−AH.

From proposition 1.5.9 we find that f0, f1, g1 are of class BCr+4 for x ∈ R3, y ∈ Rd and |ε| ≤ ε2. As g0

is linear all derivatives of order k = 1, . . . , r are bounded. Due to GA 1.3, definition 1.5.5 and (1.6.1) themaps f1 and g1 vanish for ε = 0.

Since

Df0(ξ) =

0 0 00 0 d

dhω(h)0 0 0

,


the matrix[∥

∥∂xjf0i

∥

∥

∞]

i,j=1...nis already in upper triangular form, such that we may set P := IR3 and

maxi=1...nξ∈R

n

−∂xif0i (ξ) = 0.

Therefore the first case considered in proposition 1.6.3 applies.

Choosing a suitable norm in the y–space Rd (in fact the same norm as one needs to establish the resultsgiven in section 1.4.1) it is a consequence of the fact that A is diagonalizable (cf. GA 1.2) and remark 1.6.2that the logarithmic norm of A is equal to the maximal realpart of the spectrum σ(A), hence boundedby −c0 (cf. GA 1.2). This implies

−max|y|≤

µ(

Dg0(y))

= −µ (A) ≥ c0.

Since c0 > 0 we have established the second assumption made in proposition 1.6.3. Hence we may applyproposition 1.6.3 to system (1.139).

For ε = 0 we see that the subspace

(t, ϕ, h,H) ∈ R 3+d∣

∣H = 0

is attractive (GA 1.2) and invariant with

respect to (1.111). As the map S is defined for ε = 0 and unique on R3 × BRd(), S(t, ϕ, h, 0) = 0 musttherefore hold.

In an analogous way the results found in remark 1.5.10 imply that

(t, ϕ, h,H) ∈ R 3+d∣

∣ h = 0,H = 0

isinvariant if 0 ∈ J . Hence in this situation we deduce from (1.131 e) that S(t, ϕ, 0, ε) = 0.

As for |h| < , |H| < the vector fields (1.111) and (1.139) are identical the last statement on M∣

∣

follows at once.


1.6.2 An Explicit Representation of the Attractive Invariant Manifold

The purpose of this section is to derive a sufficiently explicit representation of the map S given bycorollary 1.6.4. Expanding this map S in a Taylor series with respect to the perturbation parameter εwe give explicit formulae for the corresponding coefficient maps of the ε and ε2 terms. This is the subjectof the main result, proposition 1.6.7 of this section.

Before claiming this result let us first introduce some abbreviations:

Definition 1.6.5 Consider the vector valued maps F , G introduced in definition 1.5.5. Since F , G wereassumed to admit a representation of the form GA 1.3, we are in the position to define the mappings

• F j,02 : R3 → R via

F j,02 (t, ϕ, h) := ω(h)

ddhH(0,P(h))

(

J ∂hΦ(ϕ, h)| F j(Φ(ϕ, h), 0, t))

,

• F j,12 : R3 → L(Rd,R) via

F j,12 (t, ϕ, h)H := ω(h)

ddhH(0,P(h))

(

J ∂hΦ(ϕ, h)| ∂HF j(Φ(ϕ, h), 0, t)H)

,

• F j,22 : R3 → L(Rd × Rd,R) via

F j,22 (t, ϕ, h)(H, H) := ω(h)

ddhH(0,P(h))

(

J ∂hΦ(ϕ, h)| ∂2HF j(Φ(ϕ, h), 0, t)(

H, H)

)

,

where j = 1, 2, 3. In a most similar way we define the maps F j,03 , F j,1

3 , F j,23 (j = 1, 2, 3) by

• F j,03 : R3 → R via

F j,03 (t, ϕ, h) := 1

ddhH(0,P(h))

(

∇H(Φ(ϕ, h))∣

∣

∣F j(Φ(ϕ, h), 0, t))

,

• F j,13 : R3 → L(Rd,R) via

F j,13 (t, ϕ, h)H := 1

ddhH(0,P(h))

(

∇H(Φ(ϕ, h))∣

∣

∣∂HFj(Φ(ϕ, h), 0, t)H

)

,

• F j,23 : R3 → L(Rd × Rd,R) via

F j,23 (t, ϕ, h)(H, H) := 1

ddhH(0,P(h))

(

∇H(Φ(ϕ, h))∣

∣

∣∂2HFj(Φ(ϕ, h), 0, t)

(

H, H)

)

,

For completeness we finally set F j,01 ,F j,1

1 ,F j,21 := 0 (j = 1, 2, 3) and F0(h) =

1ω(h)0

.


Remark 1.6.6 It then follows that the m–th component (m = 1, 2, 3) of F may be written in the form

Fm(t, ϕ, h,H, ε) = F0m(h) +

∑

j=1,2,3l=0,1,2

εj F j,lm (t, ϕ, h)H[l] +O(ε4) +O(εH[3]).

Taking into account that the maps Φ and F are 2π–periodic with respect to ϕ (and t), we see that thesame must be true for the maps F j,l

m (t, ϕ, h) defined above. Thus we may consider the Fourier series ofthese maps. As shown in (1.93) the Fourier expansions of F j, Gj with respect to the time t are finite8

and more specifically,

Fm(t, ϕ, h,H, ε) = F0m(h) +

∑

j=1,2,3l=0,1,2

∑

|n|≤Nk∈Z

εj F j,lk,n,m(h)H[l] ei(kϕ+nt) +O(ε4) +O(εH[3]). (1.142)

A similar process may be carried out for the map G yielding

G(t, ϕ, h,H, ε) = AH+∑

j=1,2l=0,1

∑

|n|≤Nk∈Z

εj Gj,lk,n(h)H[l] ei(kϕ+nt) +O(ε3) +O(εH[2]). (1.143)

These representations (1.142), (1.143) will be used in section 1.6.4 as well. It now is possible to derive a

sufficiently explicit representation for the map S in terms of the quantities Gj,lk,n(h). This is the subjectof the following proposition.

Proposition 1.6.7 The map S given by corollary 1.6.4 may be written in the following form:

S(t, ϕ, h, ε) =2∑

j=1

∑

|n|≤jNk∈Z

εj Sjk,n(h) ei(kϕ+nt) + ε3 S3(t, ϕ, h, ε). (1.144)

The map S3 is 2π–periodic with respect to the variables t and ϕ and of class BCr+1. The maps Sjk,n are

of class BCr+1 as well and given by the following identities9:

S1k,n(h) = [i(k ω(h) + n)IRd −A]

−1 G1,0k,n(h)

S2k,n(h) = [i(k ω(h) + n)IRd −A]

−1(

G2,0k,n(h) +

∑

k1,k2∈Z

k1+k2=k

∑

|n1|,|n2|≤Nn1+n2=n

G1,1k1,n1

(h)S1k2,n2

(h)

−∑

k1,k2∈Z

k1+k2=k

∑

|n1|,|n2|≤Nn1+n2=n

i k1 S1k1,n1

(h)F1,0k2,n2,2

(h) + ∂hS1k1,n1

(h)F1,0k2,n2,3

(h))

.

PROOF: As the maps S ∈ BCr+4(R3 × (−ε3, ε3),Rd) are (at least) of class C4 with respect to ε, wemay write S in the form

S(t, ϕ, h, ε) =2∑

j=1

εj Sj(t, ϕ, h) + ε3 S3(t, ϕ, h, ε), (1.145)

8For simplicity let us denote the limit of the indices n arising in GA 1.3 when considering F instead of F by N again.9For the application in chapter 4 it suffices to consider the explicit formula given for S1

k,n(h).


with Sj ∈ BC 1 (j = 1, 2) where we have used (1.140 e) and

S3(t, ϕ, h, ε) := 12

1∫

0

(1 − σ)2 ∂3εS(t, ϕ, h, σ ε) dσ.

Note that since S is 2π–periodic with respect to t and ϕ, the same must be true for the functions Sj ,j = 1, 2, 3. Furthermore the boundedness of ∂3εS implies the boundedness of S3.

Consider any solution (t, ϕ, h,H) of system (1.111) contained in the invariant manifold Mε, i.e. satisfying

H(s) = S((t, ϕ, h)(s), ε) ∀s ∈ R.

Taking the derivative of this last equation with respect to the independent variable s yields

ddsH = ∂(t,ϕ,h)S(t, ϕ, h, ε)

(

dds (t, ϕ, h)

)

= ∂(t,ϕ,h)S(t, ϕ, h, ε)F(t, ϕ, h,H, ε)

= ∂(t,ϕ,h)S(t, ϕ, h, ε)F(t, ϕ, h,S(t, ϕ, h, ε), ε)

On the other hand, as H(s) solves (1.111)

ddsH = G(t, ϕ, h,S(t, ϕ, h, ε), ε) (1.146)

Using these two representations for ddsH we find the so–called equation of invariance :

∂(t,ϕ,h)S(t, ϕ, h, ε)F(t, ϕ, h,S(t, ϕ, h, ε), ε) = G(t, ϕ, h,S(t, ϕ, h, ε), ε). (1.147)

Plugging in the representation found in (1.142), (1.143) and (1.145) yields

2∑

j=1

εj ∂(t,ϕ,h)Sj(t, ϕ, h) +O(ε3)

(

F0(h) + εF1,0(t, ϕ, h) +O(ε2))

= A

2∑

j=1

εj Sj(t, ϕ, h) +O(ε3)

+2∑

j=1

∑

|n|≤Nk∈Z

εj Gj,0k,n(h) ei(kϕ+nt)

+ ε2∑

|n|≤Nk∈Z

G1,1k,n(h)S1(t, ϕ, h) ei(kϕ+nt) +O(ε3)

such that by comparing the coefficients of εj (j = 1, 2) we obtain the differential equations

∂(t,ϕ,h)S1(t, ϕ, h)F0(h) = AS1(t, ϕ, h) +∑

|n|≤Nk∈Z

G1,0k,n(0) e

i(kϕ+nt),

∂(t,ϕ,h)S2(t, ϕ, h)F0(h) = AS2(t, ϕ, h) +∑

|n|≤Nk∈Z

G2,0k,n(h) e

i(kϕ+nt)

+∑

|n|≤Nk∈Z

G1,1k,n(h)S1(t, ϕ, h) ei(kϕ+nt)

−∂(t,ϕ,h)S1(t, ϕ, h)F1,0(t, ϕ, h).


By definition of F0(h) and (1.142) this is equivalent to

∂tS1(t, ϕ, h) + ∂ϕS1(t, ϕ, h)ω(h) = AS1(t, ϕ, h) +∑

|n|≤Nk∈Z

G1,0k,n(0) e

i(kϕ+nt), (1.148)

∂tS2(t, ϕ, h) + ∂ϕS2(t, ϕ, h)ω(h) = AS2(t, ϕ, h) +∑

|n|≤Nk∈Z

G2,0k,n(h) e

i(kϕ+nt)

+∑

|n|≤Nk∈Z

G1,1k,n(h)S1(t, ϕ, h) ei(kϕ+nt)

−∂tS1(t, ϕ, h)∑

|n|≤Nk∈Z

F1k,n,1(h)e

i(kϕ+nt)

−∂ϕS1(t, ϕ, h)∑

|n|≤Nk∈Z

F1k,n,2(h)e

i(kϕ+nt)

−∂hS1(t, ϕ, h)∑

|n|≤Nk∈Z

F1k,n,3(h)e

i(kϕ+nt). (1.149)

In an analogous way as proved in lemma 1.2.3 we find the unique periodic solutions of (1.148), (1.149)to be given by

Sj(t, ϕ, h) =∑

|n|≤jNk∈Z

Sjk,n(h) ei(kϕ+nt), j = 1, 2

where Sjk,n(h) are as claimed in proposition 1.6.7. (Recall again that by assumption GA 1.2 the matrixA satisfies σ (A) ∩ iZ = ∅ such that i(k ω(h) + n)IRd −A is invertible, indeed.)

1.6.3 An Alternative Representation of the Attractive Invariant Manifold

In order to prepare considerations to follow in chapter 3, we will give an alternative representation ofthe parametrization S of the invariant manifold M (as given by corollary 1.6.4). In contrast to therepresentation (1.144) we consider an expansion with respect to the action–variable h. However, sinceall dependencies on h of the quantities which determine S are in fact dependencies on P(h), we are inthe position to express S in powers of P(h). This is the subject of the following lemma.

Lemma 1.6.8 If 0 ∈ J then the parametrization S given by corollary 1.6.4 admits a representation ofthe form

S(t, ϕ, h, ε) = P(h)S,1(t, ϕ, ε) + P(h)2 S,2(t, ϕ,P(h), ε), (1.150)

where the maps S,1, S,2 are of class Cr+2, 2π–periodic with respect to t, ϕ. Moreover S,1 satisfies thepartial differential equation

∂tS,1(t, ϕ, ε) + ∂ϕS,1(t, ϕ, ε)(

Ω0 +1ν

(

J Φ,1(ϕ)∣

∣ ∂(Q,P )F (0, 0, 0, t, ε)Φ,1(ϕ)

))

=(

A+ ∂HG(0, 0, 0, t, ε))

S,1(t, ϕ, ε) + ∂(Q,P )G(0, 0, 0, t, ε)Φ,1(ϕ) (1.151)


where Φ,1(ϕ) denotes the map defined in (1.117).

PROOF: Since 0 ∈ J we have P(0) = 0 and therefore

S(t, ϕ, h, ε) = S(t, ϕ,P−1 (P(h)) , ε) = S(t, ϕ, 0, ε) +(

S(t, ϕ,P−1 (P(h)) , ε)− S(t, ϕ,P−1 (0) , ε))

= 0 +

1∫

0

d

dσS(t, ϕ,P−1 (σ P(h)) , ε) dσ

=

1∫

0

∂hS(t, ϕ,P−1 (0) , ε) ddhP−1 (0) dσP(h)

+

1∫

0

[

∂hS(t, ϕ,P−1 (σP(h)) , ε) ddhP−1 (σP(h))− ∂hS(t, ϕ,P−1 (0) , ε) d

dhP−1 (0)]

dσ P(h)

= ∂hS(t, ϕ,P−1 (0) , ε) ddhP−1 (0) P(h)

+

1∫

0

1∫

0

d

dσ

[

∂hS(t, ϕ,P−1 (σ σP(h)) , ε) ddhP−1 (σ σP(h))

]

dσ dσP(h)

= P(h)[

∂hS(t, ϕ,P−1 (0) , ε) ddhP−1 (0)

]

+P(h)2[

1∫

0

1∫

0

∂2hS(t, ϕ,P−1 (σ σP(h)) , ε)(

ddhP−1 (σ σ P(h))

)2dσ σ dσ

+

1∫

0

1∫

0

∂hS(t, ϕ,P−1 (σ σ P(h)) , ε) d2

dh2P−1 (σ σ P(h)) dσ σ dσ]

then setting

S,1(t, ϕ, ε) := ∂hS(t, ϕ,P−1 (0) , ε) ddhP−1 (0)

S,2(t, ϕ,P(h), ε) :=

1∫

0

1∫

0

∂2hS(t, ϕ,P−1 (σ σ P(h)) , ε)(

ddhP−1 (σ σP(h))

)2dσ σ dσ

+

1∫

0

1∫

0

∂hS(t, ϕ,P−1 (σ σP(h)) , ε) d2

dh2P−1 (σ σP(h)) dσ σ dσ

yields (1.150) at once. Since S and P are Cr+4, S,1, S,2 are of class Cr+2 indeed. Substituting therepresentation (1.150) into the equations (1.115) and (1.116) yields

F2(t, ϕ, h,S(t, ϕ, h, ε), ε) = Ω0 +1ν

(

J Φ,1(ϕ)∣

∣ ∂(Q,P )F (0, 0, 0, t, ε)Φ,1(ϕ)

)

+ P(h) 1ν

(

J Φ,1(ϕ)∣

∣ ∂(Q,P )∂HF (0, 0, 0, t, ε)(

S,1(t, ϕ, ε),Φ,1(ϕ))

)

+ P(h)2 1ν

(

J Φ,1(ϕ)∣

∣ ∂(Q,P )∂HF (0, 0, 0, t, ε)(

S,2(t, ϕ,P(h), ε),Φ,1(ϕ))

)


+ 12P(h) 1

ν

(

J Φ,1(ϕ)∣

∣ ∂2(Q,P )F (0, 0, 0, t, ε)Φ,1(ϕ)[2]

)

+ P(h)2 f ,,2(ϕ,P(h),S(t, ϕ, h, ε), t, ε)(

S,1(t, ϕ, ε) + P(h)S,2(t, ϕ,P(h), ε))[2]

+ P(h)2f ,,1(ϕ,P(h),S(t, ϕ, h, ε), t, ε)

(

S,1(t, ϕ, ε) + P(h)S,2(t, ϕ,P(h), ε))

+ P(h)2 f ,,0(ϕ,P(h),S(t, ϕ, h, ε), t, ε),

as well as

F3(t, ϕ, h,S(t, ϕ, h, ε), ε) = P(h)ddhP(h)

1∂2pH(0,0)

(

H ,1(ϕ)∣

∣ ∂(Q,P )F (0, 0, 0, t, ε)Φ,1(ϕ)

)

+ P(h)2

ddhP(h)

1∂2pH(0,0)

(

H ,1(ϕ)∣

∣ ∂(Q,P )∂HF (0, 0, 0, t, ε)(S,1(t, ϕ, ε),Φ,1(ϕ)))

+ P(h)3

ddhP(h)

1∂2pH(0,0)

(

H ,1(ϕ)∣

∣ ∂(Q,P )∂HF (0, 0, 0, t, ε)(S,2(t, ϕ,P(h), ε),Φ,1(ϕ)))

+ 12

P(h)2

ddhP(h)

1∂2pH(0,0)

(

H ,1(ϕ)∣

∣ ∂2(Q,P )F (0, 0, 0, t, ε)Φ,1(ϕ)[2]

)

+ P(h)3

ddhP(h)

g,,2(ϕ,P(h),S(t, ϕ, h, ε), t, ε)(

S,1(t, ϕ, ε) + P(h)S,2(t, ϕ,P(h), ε))[2]

+ P(h)3

ddhP(h)

g,,1(ϕ,P(h),S(t, ϕ, h, ε), t, ε)(

S,1(t, ϕ, ε) + P(h)S,2(t, ϕ,P(h), ε))

+ P(h)3

ddhP(h)

g,,0(ϕ,P(h),S(t, ϕ, h, ε), t, ε).

In much the same way, (1.118) reads

G(Φ(ϕ, h),S(t, ϕ, h, ε), t, ε) = P(h)∂(Q,P )G(0, 0, 0, t, ε)Φ,1(ϕ)

+ P(h)∂HG(0, 0, 0, t, ε)S,1(t, ϕ, ε) + P(h)2∂HG(0, 0, 0, t, ε)S,2(t, ϕ,P(h), ε)

+ P(h)2h,,2(ϕ,P(h),S(t, ϕ, h, ε), t, ε)

(

S,1(t, ϕ, ε) + P(h)S,2(t, ϕ,P(h), ε))[2]

+ P(h)2h,,1(ϕ,P(h),S(t, ϕ, h, ε), t, ε)

(

S,1(t, ϕ, ε) + P(h)S,2(t, ϕ,P(h), ε))

+ P(h)2 h,,0(ϕ,P(h),S(t, ϕ, h, ε), t, ε).

Sorting these expressions by powers of P(h) yields

F2(t, ϕ, h,S(t, ϕ, h, ε), ε) = Ω0 +1ν

(

J Φ,1(ϕ)∣

∣ ∂(Q,P )F (0, 0, 0, t, ε)Φ,1(ϕ)

)

+ P(h)

[

1ν

(

J Φ,1(ϕ)∣

∣ ∂(Q,P )∂HF (0, 0, 0, t, ε)(

S,1(t, ϕ, ε),Φ,1(ϕ))

)

+ 121ν

(

J Φ,1(ϕ)∣

∣ ∂2(Q,P )F (0, 0, 0, t, ε)Φ,1(ϕ)[2]

)

]

+O(P(h)2) (1.152)

F3(t, ϕ, h,S(t, ϕ, h, ε), ε) = P(h)ddhP(h)

1∂2pH(0,0)

(

H ,1(ϕ)∣

∣ ∂(Q,P )F (0, 0, 0, t, ε)Φ,1(ϕ)

)

+ P(h)2

ddhP(h)

[

1∂2pH(0,0)

(

H ,1(ϕ)∣

∣ ∂(Q,P )∂HF (0, 0, 0, t, ε)(S,1(t, ϕ, ε),Φ,1(ϕ)))

+O(P(h)3) (1.153)


as well as

G(Φ(ϕ, h),S(t, ϕ, h, ε), t, ε) = P(h)

[

∂HG(0, 0, 0, t, ε)S,1(t, ϕ, ε) + ∂(Q,P )G(0, 0, 0, t, ε)Φ,1(ϕ)

]

+O(P(h)2). (1.154)

As shown in proposition 1.6.7, the map S satisfies the equation of invariance (1.147), which in accordanceto definition 1.5.5 may be rewritten in the form

∂tS(t, ϕ, h, ε) + ∂ϕS(t, ϕ, h, ε)F2(t, ϕ, h,S(t, ϕ, h, ε), ε)+ ∂hS(t, ϕ, h, ε)F3(t, ϕ, h,S(t, ϕ, h, ε), ε) =

AS(t, ϕ, h, ε) + G(Φ(ϕ, h),S(t, ϕ, h, ε), t, ε).

Plugging the expansion (1.150) into this last equation using (1.152)–(1.154) and comparing powers ofP(h) then yields (1.151) at once.


1.6.4 The System Restricted to the Attractive Invariant Manifold

Taking into account that Mε is globally attractive (cf. (1.140 c). of corollary 1.6.4) we see that thediscussion of system (1.111) on the invariant manifold Mε is essential for the understanding of theasymptotic behaviour. By definition of Mε, solutions of (1.111) on Mε satisfy the equation

H(s) = S((t(s), ϕ(s), h(s), ) , ε).

Hence on this manifold it suffices to consider the reduced system, i.e. the system (1.111) restricted to theattractive invariant manifold Mε :

dds (t, ϕ, h) = F(t, ϕ, h,S(t, ϕ, h, ε), ε). (1.155)

From the regularity of F , S respectively it is evident that this system is of class BCr+4. By consequenceof the representations (1.142), (1.144) the expansion with respect to ε up to order O(ε4)10 reads

ϕ = ω(h) + ε∑

F1,0k1,n1,2

(h) ei(k1ϕ+n1t)

+ ε2∑

F1,1k1,n1,2

(h)S1k2,n2

(h) ei((k1+k2)ϕ+(n1+n2)t) + ε2∑

F2,0k1,n1,2

(h) ei(k1ϕ+n1t)

+ ε3∑

(

F2,1k1,n1,2

(h)S1k2,n2

(h) + F1,1k1,n1,2

(h)S2k2,n2

(h))

ei((k1+k2)ϕ+(n1+n2)t)

+ ε3∑

F3,0k1,n1,2

(h) ei(k1ϕ+n1t) + ε4F42 (t, ϕ, h, ε)

h = ε∑

F1,0k1,n1,3

(h) ei(k1ϕ+n1t)

+ ε2∑

F1,1k1,n1,3

(h)S1k2,n2

(h) ei((k1+k2)ϕ+(n1+n2)t) + ε2∑

F2,0k1,n1,3

(h) ei(k1ϕ+n1t)

+ ε3∑

(

F2,1k1,n1,3

(h)S1k2,n2

(h) + F1,1k1,n1,3

(h)S2k2,n2

(h))

ei((k1+k2)ϕ+(n1+n2)t)

+ ε3∑

F3,0k1,n1,3

(h) ei(k1ϕ+n1t) + ε4F43 (t, ϕ, h, ε)

(1.156)

where the sums are over all |n1| , |n2| ≤ N , 2N respectively and k1, k2 ∈ Z.

In a similar way, an expansion of the reduced system with respect to P may be achieved by combining

10for the application in chapter 4 it suffices to consider the expansions of order O(ε2).


the representations (1.152) and (1.153)

ϕ = Ω0 +1ν

(

J Φ,1(ϕ)∣

∣ ∂(Q,P )F (0, 0, 0, t, ε)Φ,1(ϕ)

)

+ P(h)

(

1ν

(

J Φ,1(ϕ)∣

∣ ∂(Q,P )∂HF (0, 0, 0, t, ε)(

S,1(t, ϕ, ε),Φ,1(ϕ))

)

+ 121ν

(

J Φ,1(ϕ)∣

∣ ∂2(Q,P )F (0, 0, 0, t, ε)Φ,1(ϕ)[2]

)

)

+ P(h)2f ,2(t, ϕ,P(h), ε)

h = P(h)ddhP(h)

1∂2pH(0,0)

(

H ,1(ϕ)∣

∣ ∂(Q,P )F (0, 0, 0, t, ε)Φ,1(ϕ)

)

+ P(h)2

ddhP(h)

(

1∂2pH(0,0)

(

H ,1(ϕ)∣

∣ ∂(Q,P )∂HF (0, 0, 0, t, ε)(S,1(t, ϕ, ε),Φ,1(ϕ)))

+ 12

1∂2pH(0,0)

(

H ,1(ϕ)∣

∣ ∂2(Q,P )F (0, 0, 0, t, ε)Φ,1(ϕ)[2]

)

)

+ P(h)3

ddhP(h)

g,3(t, ϕ,P(h), ε).

(1.157)

Here we have use the representation ω(h) = Ω0 +O(P(h)2) as derived in (1.119).

Let us summarize these results in the following lemma:

Lemma 1.6.9 The reduced system (1.155) may be written in the form

ϕ = ω(h) + f(t, ϕ, h, ε)

h = g(t, ϕ, h, ε)(1.158)

and admits the epsilon / Fourier–expansion

ϕ = ω(h) +

3∑

j=1

εj∑

|n|≤3Nk∈Z

f jk,n(h) ei(kϕ+nt) + ε4 f4(t, ϕ, h, ε)

h =

3∑

j=1

εj∑

|n|≤3Nk∈Z

gjk,n(h) ei(kϕ+nt) + ε4 g4(t, ϕ, h, ε)

(1.159)

as well as the h–expansion


h = P(h)ddhP(h)

g,1(t, ϕ, ε) + P(h)2

ddhP(h)

g,2(t, ϕ, ε) + P(h)3

ddhP(h)

g,3(t, ϕ,P(h), ε),(1.160)

where the maps f j,l, gj,l are of class BCr with respect to all arguments.

In the next two chapters we will discuss systems of this general form. We complete this first chapter bystating some additional properties of the systems (1.159), (1.160) respectively.


1.6.5 Additional Properties of the Reduced System

The results given in this final section of chapter 1 will be of interest in chapter 2 where we discuss theglobal behaviour of (1.158) as well as in chapter 3 where the stability of the invariant set h = 0 (ifexisting) is discussed.

The following remark deals with the bounds of the quantitites arising on the right hand side of (1.159).

Remark 1.6.10 As we concluded above, the right hand side of (1.155) is of class BCr, r ≥ 6 and 2π–periodic with respect to t and ϕ. It is a well known result that if we write down the Fourier expansion

presented in (1.159), there exists a map g such that for every k ∈ Z, the inequality∣

∣

∣gjk,n(h)

∣

∣

∣ ,∣

∣

∣

ddhg

jk,n(h)

∣

∣

∣ ≤g(h)

max1,|k|3max1,|n|3 holds for all h ∈ R. Taking into account that the maps gjk,n are bounded, we conclude

that there exists an upper bound g∞ <∞ such that these estimates hold uniformly with respect to h, i.e.

∣

∣

∣gjk,n(h)

∣

∣

∣,∣

∣

∣

ddhg

jk,n(h)

∣

∣

∣≤ g∞

max

1, |k|3

max

1, |n|3 . (1.161)

Without loss of generality we assume that the constant g∞ is chosen sufficiently large to bound thederivatives of the map g4 up to order r as well. Finally it may be shown in a very similar way that thereexists a constant f∞ which satisfies the analogous estimates for the maps f jk,n, f

4, respectively.

In view of the application considered in chapter 4 we add the following note :

Remark 1.6.11 By consequence of the explicit representations given in (1.18), (1.90) and definition 1.6.5it follows that if F 1(q, p, 0, t) = 0 for all q, p, t ∈ R then F1,0

k1,n1,2= 0 and F1,0

k1,n1,3= 0 everywhere. In

view of the notation introduced in (1.159) this corresponds to f1k,n(h) = 0 and g1k,n(h) = 0 for all h ∈ R.

In chapter 4 we will be in the situation where remark 1.6.11 applies. Hence in this example the sums overj in (1.159) will for j = 2, 3 only. Due to this fact the discussions carried out in the following chapter 2are carried out for this slightly more special case as well (cf. the representation considered in (2.1)).

We continue with an even more explicit representation of the mappings f ,l, g,l as given in (1.157) :

Lemma 1.6.12 The maps f ,0, f ,1, g,1 and g,2 may be represented as the following Fourier polynomialsin ϕ :

f ,0(t, ϕ, ε) = 12

(

−ν ∂QFp(0, 0, 0, t, ε) + 1ν ∂P Fq(0, 0, 0, t, ε)

)

+ cos(2ϕ) 12

(

1ν ∂P Fq(0, 0, 0, t, ε) + ν ∂QFp(0, 0, 0, t, ε)

)

+ sin(2ϕ) 12

(

∂QFq(0, 0, 0, t, ε)− ∂P Fp(0, 0, 0, t, ε))

g,1(t, ϕ, ε) = 12

(

∂QFq(0, 0, 0, t, ε) + ∂P Fp(0, 0, 0, t, ε))

− cos(2ϕ) 12

(

∂QFq(0, 0, 0, t, ε)− ∂P Fp(0, 0, 0, t, ε))

+ sin(2ϕ) 12

(

1ν ∂P Fq(0, 0, 0, t, ε) + ν ∂QFp(0, 0, 0, t, ε)

)

(1.162)


f ,1(t, ϕ, ε) =

[

cos2(ϕ) 1ν ∂P∂HFq(0, 0, 0, t, ε)− sin2(ϕ) ν ∂Q∂HFp(0, 0, 0, t, ε)

+ cos(ϕ) sin(ϕ)(

∂Q∂HFq(0, 0, 0, t, ε)− ∂P∂HFp(0, 0, 0, t, ε))

]

S,1(t, ϕ, ε)

+ 12 cos3(ϕ) 1

ν ∂2P Fq(0, 0, 0, t, ε)− 1

2 sin3(ϕ) ν2 ∂2QFp(0, 0, 0, t, ε)

+ cos2(ϕ) sin(ϕ)(

∂Q∂P Fq(0, 0, 0, t, ε)− 12 ∂

2P Fp(0, 0, 0, t, ε)

)

+ ν cos(ϕ) sin2(ϕ)(

12 ∂

2QFq(0, 0, 0, t, ε)− ∂Q∂P Fp(0, 0, 0, t, ε)

)

(1.163)

g,2(t, ϕ, ε) =

[

cos2(ϕ) ∂P ∂HFp(0, 0, 0, t, ε) + sin2(ϕ) ∂Q∂HFq(0, 0, 0, t, ε)

+ cos(ϕ) sin(ϕ)(

1ν ∂P∂HFq(0, 0, 0, t, ε) + ν ∂Q∂HFp(0, 0, 0, t, ε)

)

]

S,1(t, ϕ, ε)

+ 12 cos3(ϕ) ∂2P Fp(0, 0, 0, t, ε) +

12ν sin3(ϕ) ∂2QFq(0, 0, 0, t, ε)

+ cos2(ϕ) sin(ϕ)(

ν ∂Q∂P Fp(0, 0, 0, t, ε) +12

1ν ∂

2P Fq(0, 0, 0, t, ε)

)

+ cos(ϕ) sin2(ϕ)(

12 ν

2 ∂2QFp(0, 0, 0, t, ε) + ∂Q∂P Fq(0, 0, 0, t, ε))

.

(1.164)

PROOF: As we will, for convenience, write

F (Q,P,H, t, ε) =

(

Fq(Q,P,H, t, ε)

Fp(Q,P,H, t, ε)

)

,

it then is found that for any vectors

(

xy

)

∈ R2, H ∈ Rd,

∂(Q,P )F (0, 0, 0, t, ε)

(

xy

)

=

(

x∂QFq(0, 0, 0, t, ε) + y ∂P Fq(0, 0, 0, t, ε)

x∂QFp(0, 0, 0, t, ε) + y ∂P Fp(0, 0, 0, t, ε)

)

∂(Q,P )∂HF (0, 0, 0, t, ε)

(

H,

(

xy

))

=

(

x∂Q∂HFq(0, 0, 0, t, ε) + y ∂P∂HFq(0, 0, 0, t, ε))

H(

x∂Q∂HFp(0, 0, 0, t, ε) + y ∂P∂HFp(0, 0, 0, t, ε))

H

∂2(Q,P )F (0, 0, 0, t, ε)

(

x

y

)[2]

=

(

x2 ∂2QFq(0, 0, 0, t, ε) + 2x y ∂Q∂P Fq(0, 0, 0, t, ε) + y2 ∂2

P Fq(0, 0, 0, t, ε)

x2 ∂2QFp(0, 0, 0, t, ε) + 2x y ∂Q∂P Fp(0, 0, 0, t, ε) + y2 ∂2

P Fp(0, 0, 0, t, ε)

)

.


Hence by (1.117), (1.157)

f,0(t, ϕ, ε) =

1

ν

((

cos(ϕ)−ν sin(ϕ)

)∣

∣

∣

∣

(

ν sin(ϕ) ∂QFq(0, 0, 0, t, ε) + cos(ϕ) ∂P Fq(0, 0, 0, t, ε)

ν sin(ϕ) ∂QFp(0, 0, 0, t, ε) + cos(ϕ)∂P Fp(0, 0, 0, t, ε)

))

= sin(ϕ) cos(ϕ) ∂QFq(0, 0, 0, t, ε) +1νcos2(ϕ) ∂P Fq(0, 0, 0, t, ε)

−ν sin2(ϕ) ∂QFp(0, 0, 0, t, ε)− sin(ϕ) cos(ϕ)∂P Fp(0, 0, 0, t, ε)

= cos2(ϕ) 1ν∂P Fq(0, 0, 0, t, ε)− sin2(ϕ) ν ∂QFp(0, 0, 0, t, ε)

+ 12sin(2ϕ)

(

∂QFq(0, 0, 0, t, ε)− ∂P Fp(0, 0, 0, t, ε))

= 12

(

−ν ∂QFp(0, 0, 0, t, ε) +1ν∂P Fq(0, 0, 0, t, ε)

)

+cos(2ϕ) 12

(

ν ∂QFp(0, 0, 0, t, ε) +1ν∂P Fq(0, 0, 0, t, ε)

)

+sin(2ϕ) 12

(

∂QFq(0, 0, 0, t, ε)− ∂P Fp(0, 0, 0, t, ε))

g,1(t, ϕ, ε) = 1

∂2pH(0,0)

((


)∣

∣

∣

∣

(

ν sin(ϕ) ∂QFq(0, 0, 0, t, ε) + cos(ϕ) ∂P Fq(0, 0, 0, t, ε)

ν sin(ϕ) ∂QFp(0, 0, 0, t, ε) + cos(ϕ)∂P Fp(0, 0, 0, t, ε)

))

= sin2(ϕ) ∂QFq(0, 0, 0, t, ε) +1νsin(ϕ) cos(ϕ) ∂P Fq(0, 0, 0, t, ε)

+ν sin(ϕ) cos(ϕ) ∂QFp(0, 0, 0, t, ε) + cos2(ϕ)∂P Fp(0, 0, 0, t, ε)

= 12

(

∂QFq(0, 0, 0, t, ε) + ∂P Fp(0, 0, 0, t, ε))

− cos(2ϕ) 12

(

∂QFq(0, 0, 0, t, ε)− ∂P Fp(0, 0, 0, t, ε))

+sin(2ϕ) 12

(

1ν∂P Fq(0, 0, 0, t, ε) + ν ∂QFp(0, 0, 0, t, ε)

)

and

f,1(t, ϕ, ε) =

1ν

(

(


)

∣

∣

∣

∣

∣

(

ν sin(ϕ) ∂Q∂HFq(0, 0, 0, t, ε) + cos(ϕ) ∂P∂HFq(0, 0, 0, t, ε))

S ,1(t, ϕ, ε)(

ν sin(ϕ) ∂Q∂HFp(0, 0, 0, t, ε) + cos(ϕ) ∂P∂HFp(0, 0, 0, t, ε))

S ,1(t, ϕ, ε)

)

+ 12

1ν

(

(


)

∣

∣

∣

∣

∣

. . .

. . .

(

(ν sin(ϕ))2 ∂2QFq(0, 0, 0, t, ε) + 2 ν sin(ϕ) cos(ϕ) ∂Q∂P Fq(0, 0, 0, t, ε) + cos(ϕ)2 ∂2

P Fq(0, 0, 0, t, ε)

(ν sin(ϕ))2 ∂2QFp(0, 0, 0, t, ε) + 2 ν sin(ϕ) cos(ϕ) ∂Q∂P Fp(0, 0, 0, t, ε) + cos(ϕ)2 ∂2

P Fp(0, 0, 0, t, ε)

)

)

,

g,2(t, ϕ, ε) = 1

∂2pH(0,0)

(

(


)

∣

∣

∣

∣

∣

. . .

. . .

(

ν sin(ϕ) ∂Q∂HFq(0, 0, 0, t, ε) + cos(ϕ) ∂P∂HFq(0, 0, 0, t, ε))

S ,1(t, ϕ, ε)(

ν sin(ϕ) ∂Q∂HFp(0, 0, 0, t, ε) + cos(ϕ) ∂P∂HFp(0, 0, 0, t, ε))

S ,1(t, ϕ, ε)

)

+ 12

1∂2pH(0,0)

(

(


)

∣

∣

∣

∣

∣

. . .

. . .

(

(ν sin(ϕ))2 ∂2QFq(0, 0, 0, t, ε) + 2 ν sin(ϕ) cos(ϕ) ∂Q∂P Fq(0, 0, 0, t, ε) + cos(ϕ)2 ∂2

P Fq(0, 0, 0, t, ε)

(ν sin(ϕ))2 ∂2QFp(0, 0, 0, t, ε) + 2 ν sin(ϕ) cos(ϕ) ∂Q∂P Fp(0, 0, 0, t, ε) + cos(ϕ)2 ∂2

P Fp(0, 0, 0, t, ε)

)

)

.

which eventually implies the identities (1.162).


Using the equation (1.151) asserted in lemma 1.6.8 we are in the position to establish an additional resulton the coefficient map S,1 in (1.150). More precisely we consider a particular property of 2π–periodicfunctions, introduced in the following definition.

Definition 1.6.13 Let f : R → Rn be any 2π–periodic function. Then f is called π–anti–periodic if

f(ψ + π) = −f(ψ) for all ψ ∈ R.

Given any 2π–periodic function f 6= 0 the maps

f+(ψ) :=12 (f(ψ) + f(ψ + π))

f−(ψ) :=12 (f(ψ)− f(ψ + π))

(1.165)

are called theπ–periodic, π–anti–periodic part of f respectively.

Remark 1.6.14 The proof of the following statements on π–periodic and π–anti–periodic functions isstraightforward and hence left to the reader:

1. It easily may be seen that every 2π–periodic function f may be decomposed as f = f+ + f−, wherethe π–periodic ( π–anti–periodic) part f+ (f−) of f is given by the formula (1.165), indeed.

2. Given a Fourier series f(ψ) =∑

k∈Z

fk eikψ the π–periodic, π–anti–periodic part of f are equal to

the seriesf+(ψ) =

∑

k∈Zeven

fk eikψ f−(ψ) =

∑

k∈Z

odd

fk eikψ .

3. For the pointwise multiplication of π–periodic, π–anti–periodic functions respectively the followingtable holds:

· π–periodic π–anti–periodic

π–periodic π–periodic π–anti–periodicπ–anti–periodic π–anti–periodic π–periodic

4. Let F : R → R be arbitrary, f+ a π–periodic and f− a π–anti–periodic map. Then

F (x) = F (−x) ∀x ⇒ F f+ and F f− are π–periodic

F (x) = −F (−x) ∀x ⇒ F f+ is π–periodic and F f− is π–anti–periodic.

5. The mean value 12π

2π∫

0

f−(ψ) dψ of the π–anti–periodic part of a 2π–periodic function f is zero.

In the last lemma of this chapter we finally show that by consequence of (1.151) the map S,1 and thereforeby (1.163), (1.164) the maps f ,1, g,2 are π–anti–periodic.


Lemma 1.6.15 The maps f ,1, g,2 are π–anti–periodic with respect to ϕ.

PROOF: The proof is carried out in two steps :

1. We first show that S,1 is π–anti–periodicwith respect to ϕ. Consider the identity (1.151):

∂tS,1(t, ϕ, ε) + ∂ϕS,1(t, ϕ, ε)(

Ω0 + f ,0(t, ϕ, ε))

=(

A+ ∂HG(0, 0, 0, t, ε))

S,1(t, ϕ, ε) + ∂(Q,P )G(0, 0, 0, t, ε)Φ,1(ϕ). (1.166)

As the map S describing the invariant manifold M is 2π–periodic with respect to ϕ (cf. corol-lary 1.6.4, (1.140d)), the same must be true for S,1. Thus we may decompose S,1 into a π–periodic and a π–anti–periodic part (with respect to ϕ, see remark 1.6.14) :

S,1(t, ϕ, ε) = S,1+ (t, ϕ, ε) + S,1− (t, ϕ, ε).

Plugging this representation into (1.166) we have

∂tS,1+ (t, ϕ, ε) + ∂tS,1− (t, ϕ, ε) + ∂ϕS,1+ (t, ϕ, ε)(

Ω0 + f ,0(t, ϕ, ε))

+ ∂ϕS,1− (t, ϕ, ε)(

Ω0 + f ,0(t, ϕ, ε))

=(

A+ ∂HG(0, 0, 0, t, ε))

S,1+ (t, ϕ, ε) +(

A+ ∂HG(0, 0, 0, t, ε))

S,1− (t, ϕ, ε)

+ ∂(Q,P )G(0, 0, 0, t, ε)Φ,1(ϕ). (1.167)

As the functions sin(2ϕ), cos(2ϕ) are π–periodicwith respect to ϕ, the map f ,0 is π–periodicwithrespect to ϕ (cf. (1.162)). By consequence of definition (1.117) we see immediately that on theother hand, Φ,1(ϕ) is π–anti–periodic.

As the π–periodic parts of the left and right hand side of (1.167) have to coincide, we apply thestatements given in remark 1.6.14 to compare the two corresponding quantities:

∂tS,1+ (t, ϕ, ε) + ∂ϕS,1+ (t, ϕ, ε)(

Ω0 + f ,0(t, ϕ, ε))

=(

A+ ∂HG(0, 0, 0, t, ε))

S,1+ (t, ϕ, ε). (1.168)

Since (1.168) admits a unique bounded, 2π–periodic ( with respect to t and ϕ) solution11 andS,1+ := 0 solves (1.168), we find

S,1(t, ϕ, ε) = S,1− (t, ϕ, ε),

i.e. S,1(t, ϕ, ε) is π–anti–periodic.

2. Using the representations (1.163), (1.164) of f ,1 and g,2 together with the multiplication table ofremark 1.6.14 it follows at once that f ,1 and g,2 are π–anti–periodic.

11this may be seen by applying similar arguments as used in section 4.7.2.

Chapter 2

Averaging and Passage throughResonance in Plane Systems



The aim of this chapter is to discuss systems of the form

ϕ = ω(h) +

3∑

j=2

∑

k,n∈Z

εjf jk,n(h) ei(kϕ+nt) + ε4 f4(t, ϕ, h, ε)

h =

3∑

j=2

∑

k,n∈Z

εjgjk,n(h) ei(kϕ+nt) + ε4 g4(t, ϕ, h, ε).

(2.1)

The mappings f jk,n and gjk,n are assumed to be of class BC r where r ≥ 3. In order to give a precise listof the assumptions made on (2.1) we present the setup of this chapter in a first step:

69

70 Chapter 2. Averaging and Passage through Resonance in Plane Systems



GA 2.1. ω ∈ BCr(R,R) (r ≥ 3) and there exist constants ωmin, ωmax such that

0 < ωmin ≤ ω(h) ≤ ωmax <∞ h ∈ R.

GA 2.2. Defining the set Z of relevant indices via

Z :=

(k, n) ∈ Z2∣

∣ g2k,n 6= 0 or g3k,n 6= 0

we assume that the subset R ⊂ Q of resonant frequencies, i.e.

R :=

−nk

∣

∣

∣

∣

(k, n) ∈ Z, ωmin ≤ −nk

≤ ωmax

as well as the set H := ω−1(R) are finite. More explicitely we let H admit the representation

H = hmMm=1 ⊂ R. (In case where R = H = ∅, let M := 0). We will refer to H to as the set ofresonances.

GA 2.3. The infimuminf∣

∣

∣ωmin +

n

k

∣

∣

∣,∣

∣

∣ωmax +

n

k

∣

∣

∣

∣

∣

∣(k, n) ∈ Z

is positive.

GA 2.4. lim|h|→∞

ω(h) exists and is not contained in R.

GA 2.5. ddhω(hm) 6= 0 for all 1 ≤ m ≤M

GA 2.6. There exists a constant b∞ such that the estimates

∣

∣

∣fjk,n(h)

∣

∣

∣ ,∣

∣

∣

ddhf

jk,n(h)

∣

∣

∣ ,∣

∣

∣gjk,n(h)

∣

∣

∣ ,∣

∣

∣

ddhg

jk,n(h)

∣

∣

∣ ≤ b∞

max

1, |k|3

max

1, |n|3 (2.2)

are fulfilled for all h ∈ R. Moreover we assume without loss of generality that the maps f4, g4 arebounded uniformly by b∞.

Remark 2.1.1 Following the statements given in remark 1.6.9 and remark 1.6.11, the reduced systemderived in chapter 1 is of the form (2.1) (where f jk,n := 0, gjk,n := 0 for |n| > 3N), provided that

F 1(q, p, 0, t) = 0 identically.As the sum over the index n in (1.159) is finite, it is a simple consequence of the definition of R thatfor the reduced system (1.159) the set R of resonant frequencies is finite. This bound for n implies theproperty assumed in GA 2.3 as well. Moreover we conclude from remark 1.6.10 that the reduced system(1.159) fulfills GA 2.6.

Hence the reduced system satifies GA2 provided that the maps Ω, P of chapter 1 are appropriate to satisfythe additional assumptions made in GA2.

2.2. Near Identity Transformations, Small Denominators and Averaging 71

2.2 Near Identity Transformations, Small Denominators and

Averaging

In this section we will discuss the possibility to apply near identity transformations on the action variableh such that the resulting representation of (2.1) is easier to discuss qualitatively. We will see that it ispossible to remove the Fourier coefficient maps gjk,n for all ”frequencies” −n

k up to at most one resonantfrequency. Due to small denominators in the transformations applied, it will not be possible to removethe terms corresponding to this resonant frequency on the entire domain h ∈ R but only outside anO(ε)–neighbourhood of the corresponding resonance.

The process carried out to remove the non–resonant terms is inspired by the standard way of averaging.However we will not drop higher order terms but consider the entire transformed system. Hence we willnot have to discuss the error made by approximating the original system by the averaged system (seee.g. section 11.3.1 in [11]). On the other hand, it will be necessary to work out these terms of the vectorfield which determine the qualitative behaviour and to control the size of the remaining ”perturbation”terms.

2.2.1 On Resonances and Small Denominators

In this first section we define the notion of resonance and prepare the procedure of averaging by givingcrucial estimates on small denominators.

Lemma 2.2.1 Define the distance d(h) between the frequency ω(h) and the set R of resonant frequenciesby

d(h) := dist(ω(h),R) = min1≤m≤M

|ω(h)− ω(hm)| . (2.3)

Then d ∈ C(R,R) and d(h) = 0 for all h ∈ H. Moreover there exists a constant c1 ∈ (0, 1] such that

d(h) ≥ c1 min1≤m≤M

1, |h− hm| (2.4)

for all h ∈ R.

h

ωmax

ωmin

ω( )h

hm

m

hd(h)

ω( )

ω( )ωmax

h

m

ω( )

h

hm

minω

h

d(h)

Figure 2.1: Two examples for the map ω together with sets R, H and the plot of d(h).


PROOF: Since the absolute value is a continuous map, it follows at once by definition of d that d ∈C(R,R) and d(h) = 0 for all h ∈ R. It therefore remains to prove (2.4).

Recall that by GA 2.5 ddhω(hm) 6= 0 such that b0 := min

1≤m≤M

∣

∣

ddhω(hm)

∣

∣ is positive. Moreover it is assumed

in GA 2.1 that b1 := max

1, suph∈R

∣

∣

∣

d2

dh2ω(h)∣

∣

∣

is finite. Since H is maximal and finite and lim|h|→∞

ω(h) 6∈ R(GA 2.4) we conclude that d(h) = 0 ⇔ h ∈ H and lim

|h|→∞d(h) 6= 0. We proceed in the following three

steps:

1. The map d(h) is bounded from below for large h, i.e. there exist positive constants b2 and b3 ≥ b0/b1such that

d(h) = min1≤m≤M

|ω(h)− ω(hm)| ≥ b2 ∀ |h| ≥ b3.

2. The set I :=

h ∈ R

∣

∣

∣

∣

min1≤m≤M

|h− hm| ≥ b0b1, |h| ≤ b3

is compact and contains no zeroes of the

function d. Thus the continuous map d∣

∣

∣

Iis bounded uniformly from below by some constant b4 > 0:

d(h) ≥ b4 ∀h ∈ I.

3. Given any |h| ≤ b3 with min1≤m≤M

|h− hm| ≤ b0b1, there exists an integer 1 ≤ m ≤M such that

d(h) = |ω(h)− ω(hm)| .

Then∣

∣ω(h)− ω(hm)− ddhω(hm) (h− hm)

∣

∣ ≤ 12b1 |h− hm|2 ≤ 1

2b0 |h− hm| ,

hence

d(h) = |ω(h)− ω(hm)|≥

∣

∣

∣

∣

ddhω(hm) (h− hm)

∣

∣−∣

∣ω(h)− ω(hm)− ddhω(hm) (h− hm)

∣

∣

∣

∣

≥∣

∣

ddhω(hm)

∣

∣ |h− hm| − 12b0 |h− hm|

≥ 12b0 |h− hm|

≥ 12b0 min

1≤m≤M|h− hm| .

Summarizing the estimates found in these three cases we complete the proof by setting

c1 := min

1, 12b0, b2, b4

.

In the next lemma we give some important bounds for particular denominators appearing in what follows.


Lemma 2.2.2 There exist constants ∈ (0, 1] , c2 ≥ 1 such that the open balls BR(hm, ), 1 ≤ m ≤Mare disjoint and for every (k, n) ∈ Z (k 6= 0) the following estimates are true:

2.5 a. If −nk 6∈ R, h ∈ R, then

1

|k ω(h) + n| ≤ c2

2.5 b. If −nk ∈ R, h ∈ R \ H, then

1

|k ω(h) + n| ≤c2d(h)

2.5 c. If for 1 ≤ m ≤M fixed, −nk ∈ R \ ω(hm) and h ∈ BR(hm, ) then

1

|k ω(h) + n| ≤ c2.

(i.e. if h is near a resonance hm and −nk

is a resonant frequency corresponding to a different resonance

then the denominator |k ω(h) + n| is bounded uniformly from below.)

PROOF:

a) Since −nk 6∈ R it follows from the definition of R that either of the following two cases apply:

1.) −nk < ωmin: using GA 2.3 we have

∣

∣

∣ω(h) +n

k

∣

∣

∣ =

∣

∣

∣

∣

ω(h)− −nk

∣

∣

∣

∣

≥ ωmin −−nk

=∣

∣

∣ωmin +n

k

∣

∣

∣

≥ inf∣

∣

∣ωmin +n

k

∣

∣

∣ ,∣

∣

∣ωmax +n

k

∣

∣

∣

∣

∣

∣ (k, n) ∈ Z

=: b0 > 0

and hence

|k ω(h) + n| = |k|∣

∣

∣ω(h) +

n

k

∣

∣

∣≥∣

∣

∣ω(h) +

n

k

∣

∣

∣≥ b0.

2.) ωmax <−nk : in a very similar way we find

|k ω(h) + n| ≥∣

∣

∣ω(h) +

n

k

∣

∣

∣=

−nk

− ω(h) ≥ −nk

− ωmax =∣

∣

∣ωmax +

n

k

∣

∣

∣≥ b0.

b) If −nk ∈ R then there exists hm ∈ H such that −n

k = ω(hm), hence

|k ω(h) + n| = |k|∣

∣

∣ω(h) +

n

k

∣

∣

∣= |k| |ω(h)− ω(hm)| ≥ d(h).

By assumption h 6∈ H hence d(h) 6= 0 which implies

1

|k ω(h) + n| ≤1

d(h).

c) Setting˜ := 1

3 min |hm − hm| |1 ≤ m, m ≤M,m 6= m

the open balls BR(hm, ˜) are disjoint. Using this quantity ˜ we define

1

b3:= min |q − q| | q, q ∈ R, q 6= q

:= min

1, ˜,1

2 b3 max∣

∣

ddhω(h)

∣

∣

∣

∣ h ∈ BR(hm, ˜), 1 ≤ m ≤M

.


For 1 ≤ m ≤M fixed, let km, nm be integers such that ω(hm) = −nmkm

and write

|k ω(h) + n| = |k|∣

∣

∣ω(h) +n

k

∣

∣

∣ ≥∣

∣

∣

∣

ω(h)− ω(hm) +−nmkm

+n

k

∣

∣

∣

∣

≥∣

∣

∣

∣

∣

∣

∣

∣

−nmkm

− −nk

∣

∣

∣

∣

− |ω(h)− ω(hm)|∣

∣

∣

∣

.

(2.6)

Let −nk be as assumed, then −n

k 6= −nmkm

. Moreover as −nk ,

−nmkm

∈ R it follows immediately that

∣

∣

∣

∣

−nmkm

− −nk

∣

∣

∣

∣

≥ 1

b3.

Given any h ∈ BR(hm, ) ⊂ BR(hm, ˜) we find

|ω(h)− ω(hm)| ≤ max∣

∣

ddhω(h)

∣

∣

∣

∣ h ∈ BR(hm, ˜)

|h− hm|≤ max

∣

∣

ddhω(h)

∣

∣

∣

∣ h ∈ BR(hm, ˜), 1 ≤ m ≤M

≤ 1

2 b3.

From (2.6) we therefore conclude

|k ω(h) + n| ≥∣

∣

∣

∣

−nmkm

− −nk

∣

∣

∣

∣

− |ω(h)− ω(hm)| ≥ 1

2 b3.

We complete the proof of lemma 2.2.2 by setting

c2 := max 1/b0, 1, 2 b3 .

We continue the preparations by proving the following result on the existence and boundedness of theh–dependent Fourier series being used in the definition of the transformations applied below.

Lemma 2.2.3 There exists a constant c3 > 0 such that for any J ⊂ Z2 the following bounds hold :

∑

(k,n)∈J

∣

∣

∣gjk,n(h)

∣

∣

∣ ,∑

(k,n)∈J

∣

∣

∣

ddhg

jk,n(h)

∣

∣

∣

∑

(k,n)∈J

∣

∣

∣n gjk,n(h)

∣

∣

∣ ,∑

(k,n)∈J

∣

∣

∣k gjk,n(h)

∣

∣

∣

≤ c3 (2.7)

for j = 2, 3.

PROOF: Set

c3 :=∑

k,n∈Z

b∞max 1, k2max 1, n2 (2.8)


and apply GA 2.6 :∑

(k,n)∈J

∣

∣

∣

dk

dhkgjk,n(h)

∣

∣

∣ ≤∑

(k,n)∈J

b∞

max

1, |k|3

max

1, |n|3 ≤ c3 for k = 0, 1. In a

similar way,∑

(k,n)∈J

∣

∣

∣n gjk,n(h)

∣

∣

∣≤

∑

(k,n)∈J

b∞

max

1, |k|3

max 1, n2≤ c3 and eventually

∑

(k,n)∈J

∣

∣

∣k gjk,n(h)

∣

∣

∣≤ c3.

Lemma 2.2.4 Let I ⊂ R be open, J ⊂ Z and b > 0 be given such that for every (k, n) ∈ J the maph 7→ 1

k ω(h)+n is bounded uniformly:

1

|k ω(h) + n| ≤ b ∀h ∈ I.

Then the maps uj defined by

uj(t, ϕ, h) :=∑

(k,n)∈J

−gjk,n(h)i(k ω(h) + n)

ei(kϕ+nt) j = 2, 3 (2.9)

satisfy the estimates:

2.10 a.∣

∣uj(t, ϕ, h)∣

∣ ,∣

∣∂tuj(t, ϕ, h)

∣

∣ ,∣

∣∂ϕuj(t, ϕ, h)

∣

∣ ≤ c4 b

2.10 b.∣

∣∂huj(t, ϕ, h)

∣

∣ ≤ c4 b (1 + b)

uniformly with respect to t, ϕ ∈ R and h ∈ I, where c4 := c3 max

1, suph∈R

∣

∣

ddhω(h)

∣

∣

.

PROOF: Defining c4 as in the claim it follows from the assumptions together with the estimates (2.7)given in lemma 2.2.3 that for j = 2, 3

∣

∣uj(t, ϕ, h)∣

∣ =

∣

∣

∣

∣

∣

∣

∑

(k,n)∈J


ei(kϕ+nt)

∣

∣

∣

∣

∣

∣

≤∑

(k,n)∈J

∣

∣

∣

∣

∣

gjk,n(h)

i(k ω(h) + n)

∣

∣

∣

∣

∣

≤∑

(k,n)∈Zb∣

∣

∣gjk,n(h)

∣

∣

∣ ≤ c4 b.

Using the formal series

∂tuj(t, ϕ, h) =

∑

(k,n)∈Jin


ei(kϕ+nt)

∂ϕuj(t, ϕ, h) =

∑

(k,n)∈Jik


ei(kϕ+nt)


we show in a very similar way that ∂tuj and ∂ϕu

j exist and

∣

∣∂tuj(t, ϕ, h)

∣

∣ ,∣

∣∂ϕuj(t, ϕ, h)

∣

∣ ≤ c4 b.

This proves (2.10 a). In order to establish (2.10 b) we first recall that by GA 2.1 suph∈R

∣

∣

ddhω(h)

∣

∣ is finite.

Taking the derivative of (2.9) with respect to h yields

∂huj(t, ϕ, h) =

∑

(k,n)∈J

(

− ddhg

jk,n(h)

i(k ω(h) + n)+ ik d

dhω(h)gjk,n(h)

(i(k ω(h) + n))2

)

ei(kϕ+nt).

Using GA 2.6 and lemma 2.2.3, the same estimates carried out for∣

∣uj(t, ϕ, h)∣

∣ lead to

∣

∣

∣

∣

∣

∣

∑

(k,n)∈J

− ddhg

jk,n(h)

i(k ω(h) + n)ei(kϕ+nt)

∣

∣

∣

∣

∣

∣

≤ c4 b.

The second series may be bounded as follows:∣

∣

∣

∣

∣

∣

∑

(k,n)∈Ji k d

dhω(h)gjk,n(h)

(i(k ω(h) + n))2 e

i(kϕ+nt)

∣

∣

∣

∣

∣

∣

≤ suph∈R

∣

∣

ddhω(h)

∣

∣

∑

(k,n)∈J

∣

∣

∣i k gjk,n(h)

∣

∣

∣ b2

≤ suph∈R

∣

∣

ddhω(h)

∣

∣ c3 b2

and hence∣

∣∂huj(t, ϕ, h)

∣

∣ ≤ c4 b+ c4 b2 ≤ c4 b (1 + b)

as claimed.

We close this section with a general result implied by the inverse mapping theorem.

Lemma 2.2.5 Consider a finite union I =

L⋃

l=1

Il of open intervals Il ⊂ R where the closures Il are

disjoint. Define

cI :=13 min1≤l,l≤Ll 6=l

dist(Il, Il) (2.11)

(in the case L = 1 set cI := 1). Assume that we are given a map u ∈ C1(R2 × I,R) such that

|u(t, ϕ, h)| ≤ cI , |∂hu(t, ϕ, h)| ≤ 12 ∀(t, ϕ, h) ∈ R2 × I

and assume that u is 2π–periodic with respect to t and ϕ. Let U ⊂ R3 denote the image of the mapping

R2 × I −→ R3 : (t, ϕ, h) 7→ (t, ϕ, h+ u(t, ϕ, h)) . (2.12)

Then there exists a map v ∈ C1(U ,R) such that the following assertions hold:


2.13 a. v is 2π–periodic with respect to t and ϕ and bounded by u∞ := sup |u(t, ϕ, h)| | t, ϕ ∈ R, h ∈ I.

2.13 b. For every (t, ϕ, h) ∈ R2 × I, the identity

h = h+ u(t, ϕ, h) + v(t, ϕ, h+ u(t, ϕ, h))

holds.

h

cI

h

c

cI

I

cI

u

v

8u

hh

U

I

cI

Figure 2.2: Illustration of the situation discussed in lemma 2.2.5 (where (t, ϕ) are fixed)

Note that setting h := h + u(t, ϕ, h), the statement (2.13b) reads h = h + v(t, ϕ, h). Hence for (t, ϕ)fixed, v defines the inverse mapping of h+ u(t, ϕ, h) (cf. figure 2.2). From this point of view it is evidentthat applying the inverse mapping theorem leads to the assertion of lemma 2.2.5 in a routine manner.

The statement given in lemma 2.2.5 is a general result and will ensure that the change of coordinatesintroduced in the next section is well defined. Taking into account the possibility of small denominators,we will see that the crucial point of averaging consist in verifying the assumptions of lemma 2.2.5.


2.2.2 The Application of Particular Near Identity Transformations

We now are in the position to introduce the change of coordinates announced and to calculate thetransformed vector field. The following lemma summarizes the results found in section 2.2.1 and providesthe tools needed to prove an important result of this chapter given in proposition 2.2.7.

Lemma 2.2.6 Assume we are given an integer p ∈ 0, 1, a constant c5 > 0, a set J ⊂ Z and a familyIε,δ of subsets of R where (ε, δ) are in a subset of [−1, 1]× R∗

+ and assume that there exists a functionb(ε, δ) such that the following statements are true:

2.14 a. For all (k, n) ∈ J , h ∈ Iε,δ the estimate 1|k ω(h)+n| ≤ b(ε, δ) applies.

2.14 b. Iε,δ is a finite union of open intervals Il,ε,δ, l = 1, . . . , L where the closures Il,ε,δ are disjoint.For cI(ε, δ) as defined in (2.11) the estimates 2 ε2 c4 b(ε, δ) ≤ cI(ε, δ) and2 ε2 c4 b(ε, δ) (1 + b(ε, δ)) ≤ 1

2 are true.

2.14 c. The quantities(

|ε|δ

)2 (1−p)b(ε, δ) (1 + b(ε, δ)),

(

|ε|δ

)1−pb(ε, δ) and δ

(

|ε|δ

)p

are bounded by c5.

Then the map u(t, ϕ, h, ε) := ε2 u2(t, ϕ, h) + ε3u3(t, ϕ, h) where

uj(t, ϕ, h) :=∑

(k,n)∈J


ei(kϕ+nt) j = 2, 3 (2.15)

satisfies all assumptions made in lemma 2.2.5. The near identity change of coordinates h = h+u(t, ϕ, h, ε)defined as in (2.12) transforms (2.1) into the system

ϕ = ω(h+ v(t, ϕ, h, ε)) +

3∑

j=2

∑

k,n∈Z

εjf jk,n(h) ei(kϕ+nt) + ε3+p δ1−p f3(t, ϕ, h, ε, δ) + ε4 f4(t, ϕ, h, ε)

˙h =3∑

j=2

∑

(k,n)∈Jcεjgjk,n(h) e

i(kϕ+nt) + ε2 (1+p) δ2 (1−p) g2(t, ϕ, h, ε, δ)

+ ε3+p δ1−p g3(t, ϕ, h, ε, δ) + ε4 g4(t, ϕ, h, ε),

(2.16)

where v is the map satisfying h = h+ v(t, ϕ, h, ε) (cf. lemma 2.2.5) and Jc = (k, n) ∈ Z | (k, n) 6∈ J.

The functions f jk,n, fj, gjk,n and gj are bounded by a constant B∞(b∞, c3, c4, c5), uniformly with respect

to(

t, ϕ, h)

∈ U .

Finally, the following special cases apply:

2.17 a. if f2k,n = 0 for all (k, n) ∈ Z2 then f3 = 0

2.17 b. if g2k,n = 0 for all (k, n) ∈ J then g2 = 0.


PROOF: It is evident that for every 1 ≤ l ≤ L, lemma 2.2.4 may be applied on the open interval Il,ε,δ.Hence for the maps uj defined by (2.15) the estimate given in (2.10 a) applies. Consequently

|u(t, ϕ, h, ε)| ≤ ε2∣

∣u2(t, ϕ, h)∣

∣+ |ε|3∣

∣u3(t, ϕ, h)∣

∣ ≤ ε2 c4 b(ε, δ) + |ε|3 c4 b(ε, δ)≤ 2 ε2 c4 b(ε, δ) =: u∞(ε, δ).

(2.18)

By assumption (2.14 b) we then have u∞(ε, δ) ≤ cI(ε, δ). In a very similar way (2.10 b) implies

|∂hu(t, ϕ, h, ε)| ≤ 2 ε2 c4 b(ε, δ) (1 + b(ε, δ)) ≤ 12 .

This establishes the assumptions made in lemma 2.2.5. By consequence of this lemma the map definedin (2.12) is bijective and therefore defines a change of coordinates:

h = h+3∑

j=2

εj uj(t, ϕ, h). (2.19)

Taking the derivative with respect to t we obtain

˙h = h+

3∑

j=2

εj[

∂tuj(t, ϕ, h) + ∂ϕu

j(t, ϕ, h) ϕ+ ∂huj(t, ϕ, h) h

]

= h− ε4 g4(t, ϕ, h, ε) +

3∑

j=2

εj[


j(t, ϕ, h)ω(h)]

+

3∑

j=2

εj[

∂ϕuj(t, ϕ, h) (ϕ − ω(h)) + ∂hu

j(t, ϕ, h) h]

+ ε4 g4(t, ϕ, h, ε). (2.20)

By definition (2.15) of uj we find


j(t, ϕ, h)ω(h) =∑

(k,n)∈Ji(k ω(h) + n)


ei(kϕ+nt)

=∑

(k,n)∈J−gjk,n(h) ei(kϕ+nt)

such that plugging in the equation for h in (2.1) yields

h− ε4 g4(t, ϕ, h, ε) +

3∑

j=2

εj[


j(t, ϕ, h)ω(h)]

=

3∑

j=2

∑

(k,n)∈Zεj gjk,n(h) e

i(kϕ+nt) +

3∑

j=2

εj∑

(k,n)∈J−gjk,n(h) ei(kϕ+nt)

=

3∑

j=2

∑

(k,n)∈Jcεj gjk,n(h) e

i(kϕ+nt).


Plugging this result into (2.20) then yields

˙h =

3∑

j=2

∑


i(kϕ+nt) +

3∑

j=2

εj[

∂ϕuj(t, ϕ, h) (ϕ − ω(h)) + ∂hu

j(t, ϕ, h) h]

+ε4 g4(t, ϕ, h, ε)

=

3∑

j=2

∑


i(kϕ+nt) +

3∑

j=2

∑

(k,n)∈Jcεj(

gjk,n(h)− gjk,n(h))

ei(kϕ+nt)

+3∑

j=2

εj[

∂ϕuj(t, ϕ, h) (ϕ− ω(h)) + ∂hu

j(t, ϕ, h) h]

+ε4 g4(t, ϕ, h, ε).

=

3∑

j=2

∑


i(kϕ+nt) (2.21)

+ε2 (1+p) δ2 (1−p)[

1

ε2p δ2 (1−p) ∂hu2(t, ϕ, h) h

]

+ε3+p δ1−p[

1

εp δ1−p∂hu

3(t, ϕ, h) h+1

ε1+p δ1−p∂ϕu

2(t, ϕ, h) (ϕ− ω(h))

+1

ε1+p δ1−p

∑

(k,n)∈Jc

(

g2k,n(h)− g2k,n(h))

ei(kϕ+nt)

]

+ε4

[

1

ε∂ϕu

3(t, ϕ, h) (ϕ− ω(h)) +1

ε

∑

(k,n)∈Jc

(


ei(kϕ+nt) + g4(t, ϕ, h, ε)

]

.

Let v be the map given by lemma 2.2.5 such that h = h + u(t, ϕ, h, ε) + v(t, ϕ, h + u(t, ϕ, h), ε), henceh = h+ v(t, ϕ, h) for every

(

t, ϕ, h)

∈ U . We then define the abbreviations

g2(t, ϕ, h, ε, δ) :=1

ε2p δ2 (1−p) ∂hu2(t, ϕ, h) h

g3(t, ϕ, h, ε, δ) :=1

εp δ1−p∂hu

3(t, ϕ, h) h+1

ε1+p δ1−p∂ϕu

2(t, ϕ, h) (ϕ− ω(h))

+1

ε1+p δ1−p

∑

(k,n)∈Jc

(


ei(kϕ+nt)

g4(t, ϕ, h, ε) :=1

ε∂ϕu

3(t, ϕ, h) (ϕ − ω(h)) +1

ε

∑

(k,n)∈Jc

(


ei(kϕ+nt) + g4(t, ϕ, h, ε).

In this definition, the expressions ϕ, h must be substituted according to the identities given in (2.1) andh has to be replaced by h = h+ v(t, ϕ, h).


Then (2.21) simplifies to

˙h =

3∑

j=2

∑


i(kϕ+nt) + ε2 (1+p) δ2 (1−p) g2(t, ϕ, h, ε, δ)

+ε3+p δ1−p g3(t, ϕ, h, ε, δ) + ε4 g4(t, ϕ, h, ε).

Note that by definitions of u2, g2 respectively the statement given in (2.17b) follows at once.

In a next step we prove that the maps g2, g3 and g4 are bounded uniformly. We substitute ϕ, h using(2.1) in definition (2.2.2):

g2(t, ϕ, h, ε, δ) =1

ε2p δ2 (1−p) ∂hu2(t, ϕ, h)

3∑

j=2

∑

(k,n)∈Zεjgjk,n(h) e

i(kϕ+nt) + ε4 g4(t, ϕ, h, ε)

g3(t, ϕ, h, ε, δ) =1

εp δ1−p∂hu

3(t, ϕ, h)

3∑

j=2

∑

(k,n)∈Zεjgjk,n(h) e

i(kϕ+nt) + ε4 g4(t, ϕ, h, ε)

+1

ε1+p δ1−p∂ϕu

2(t, ϕ, h)

3∑

j=2

∑

(k,n)∈Zεjf jk,n(h) e

i(kϕ+nt) + ε4 f4(t, ϕ, h, ε)

+1

ε1+p δ1−p

∑

(k,n)∈Jc

(


ei(kϕ+nt)

g4(t, ϕ, h, ε) =1

ε∂ϕu

3(t, ϕ, h)

3∑

j=2

∑

(k,n)∈Zεjf jk,n(h) e

i(kϕ+nt) + ε4 f4(t, ϕ, h, ε)

+1

ε

∑

(k,n)∈Jc

(


ei(kϕ+nt) + g4(t, ϕ, h, ε).

Taking into account that u∞(ε, δ) is a bound of v (cf. (2.13 a)), we deduce

∣

∣

∣

∣

∣

∣

∑

(k,n)∈Jc

(

gjk,n(h)− gjk,n(h))

ei(kϕ+nt)

∣

∣

∣

∣

∣

∣

≤∑

(k,n)∈Jcsuph∈R

∣

∣

∣

ddhg

jk,n(h)

∣

∣

∣

∣

∣h− h∣

∣ ≤ c3∣

∣h− h∣

∣

= c3∣

∣v(t, ϕ, h)∣

∣ ≤ c3 u∞(ε, δ)


for j = 2, 3. This enables us to find a bound for g2. Using lemma 2.2.3 and lemma 2.2.4 together with(2.14 c) we have

∣

∣g2(t, ϕ, h, ε, δ)∣

∣ ≤ ε2∣

∣

∣

∣

1

ε2p δ2 (1−p) ∂hu2(t, ϕ, h)

∣

∣

∣

∣

3∑

j=2

∣

∣

∣

∣

∣

∣

∑

(k,n)∈Zgjk,n(h) e

i(kϕ+nt)

∣

∣

∣

∣

∣

∣

+∣

∣g4(t, ϕ, h, ε)∣

∣

≤ ε21

ε2p δ2 (1−p) c4 b(ε, δ) (1 + b(ε, δ)) (2 c3 + b∞)

=

( |ε|δ

)2 (1−p)b(ε, δ) (1 + b(ε, δ)) (2 c3 + b∞) c4

≤ c5 (2 c3 + b∞) c4 =: b1.

(Recall that b∞ denotes the bound of f4, g4 in GA 2.6). In an analogous way we determine a bound forg3 as well:

∣

∣g3(t, ϕ, h, ε, δ)∣

∣ ≤ ε2∣

∣

∣

∣

1

εp δ1−p∂hu

3(t, ϕ, h)

∣

∣

∣

∣

3∑

j=2

∣

∣

∣

∣

∣

∣

∑

(k,n)∈Zgjk,n(h) e

i(kϕ+nt)

∣

∣

∣

∣

∣

∣

+∣

∣g4(t, ϕ, h, ε)∣

∣

+ε2∣

∣

∣

∣

1

ε1+p δ1−p∂ϕu

2(t, ϕ, h)

∣

∣

∣

∣

3∑

j=2

∣

∣

∣

∣

∣

∣

∑

k,n∈Z

f jk,n(h) ei(kϕ+nt)

∣

∣

∣

∣

∣

∣

+∣

∣f4(t, ϕ, h, ε)∣

∣

+

∣

∣

∣

∣

1

ε1+p δ1−p

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∑

(k,n)∈Jc

(


ei(kϕ+nt)

∣

∣

∣

∣

∣

∣

≤ ε2∣

∣

∣

∣

1

εp δ1−p

∣

∣

∣

∣

c4 b(ε, δ) (1 + b(ε, δ)) (2 c3 + b∞) + ε2∣

∣

∣

∣

1

ε1+p δ1−p

∣

∣

∣

∣

c4 b(ε, δ) (2 c3 + b∞)

+

∣

∣

∣

∣

1

ε1+p δ1−p

∣

∣

∣

∣

c3 u∞(ε, δ)

= δ

( |ε|δ

)p(

( |ε|δ

)2 (1−p)b(ε, δ) (1 + b(ε, δ))

)

(2 c3 + b∞) c4

+

( |ε|δ

)1−pb(ε, δ) ((2 c3 + b∞) c4 + 2 c3 c4)

≤ (c5)2(2 c3 + b∞) c4 + c5 ((2 c3 + b∞) c4 + 2 c3 c4) =: b2.

Finally, a bound for g4 is obtained as follows:

∣

∣g4(t, ϕ, h, ε)∣

∣ ≤ ε2∣

∣

∣

∣

1

ε∂ϕu

3(t, ϕ, h)

∣

∣

∣

∣

3∑

j=2

∣

∣

∣

∣

∣

∣

∑

k,n∈Z

f jk,n(h) ei(kϕ+nt)

∣

∣

∣

∣

∣

∣

+∣

∣f4(t, ϕ, h, ε)∣

∣

+1

|ε|

∣

∣

∣

∣

∣

∣

∑

(k,n)∈Jc

(


ei(kϕ+nt)

∣

∣

∣

∣

∣

∣

+∣

∣g4(t, ϕ, h, ε)∣

∣

≤ |ε| c4 b(ε, δ) (2 c3 + b∞) +1

|ε| c3 u∞(ε, δ) + b∞


≤ δ

( |ε|δ

)p(

( |ε|δ

)1−pb(ε, δ)

)

((2 c3 + b∞) c4 + 2 c3 c4) + b∞

≤ (c5)2((2 c3 + b∞) c4 + 2 c3 c4) + b∞ =: b3.

Therefore the maps gjk,n and gj are bounded uniformly with respect to(

t, ϕ, h)

∈ U by the constantG∞ := max b1, b2, b3, b∞.

The statement on the ϕ–equation and the corresponding transformed right hand side is proved in a similarway:

ϕ = ω(h) +

3∑

j=2

∑

k,n∈Z

εjf jk,n(h) ei(kϕ+nt)

+

3∑

j=2

∑

k,n∈Z

εj(f jk,n(h)− f jk,n(h)) ei(kϕ+nt) + ε4 f4(t, ϕ, h, ε)

= ω(h+ v(t, ϕ, h, ε)) +

3∑

j=2

∑

k,n∈Z

εjf jk,n(h) ei(kϕ+nt)

+ε3+p δ1−pf3(t, ϕ, h, ε, δ) + ε4 f4(t, ϕ, h, ε)

where

f3(t, ϕ, h, ε, δ) :=1

ε1+p δ1−p

∑

k,n∈Z

(f2k,n(h)− f2

k,n(h)) ei(kϕ+nt)

f4(t, ϕ, h, ε) :=1

ε

∑

k,n∈Z

(f3k,n(h)− f3

k,n(h)) ei(kϕ+nt) + f4(t, ϕ, h, ε).

One then shows again that f jk,n and f j are bounded uniformly by a constant F∞. In view of the definition

of f3 we eventually find the statement given in (2.17 a) to be true.

We complete the proof by setting B∞ := max F∞, G∞.


2.2.3 Splitting the System into Inner and Outer Regions

We apply the result deduced in the preceeding section for appropriate choice of the sets I, J . Thefollowing proposition shows that the majority of the terms on the right hand side of (2.1) have noinfluence on the qualitative behaviour. Depending on the range of h considered, it suffices to focus on the”constant” Fourier coefficients gj0,0 and, O(ε)–close to the resonances, in addition the Fourier coefficientscorresponding to the resonant frequencies. The influence of the remaining terms may be controlled bychoosing an appropriate size for the neighbourhood of the resonance considered.

We will split the h–axis into O(1)–neighbourhoods of the resonances (i.e. the ”Inner Regions”) and theremaining regions, reaching O(ε)–close to the resonances (i.e. the ”Outer Regions”). We emphasize thatthe regions considered overlap. This is achieved by choosing the parameter δ (determining the size of theO(ε)–neighbourhoods) appropriately.

Proposition 2.2.7 The following statements are true:

1. Averaging on the ”outer region”

There exists a constant δ∞ > 0 such that for any 0 < δ ≤ δ∞, there is εO(δ) > 0 satisfying thefollowing statement:

Choosing |ε| ≤ εO(δ) and setting

IO

ε,δ = R \M⋃

m=1

BR(hm,|ε|δ ), JO = Z \ (0, 0) (2.22)

the assumptions of lemma 2.2.6 are satisfied with p = 0, Iε,δ = IO

ε,δ and J = JO. Thus thetransformation defined in (2.12) may be applied to (2.1). This yields the transformed system

ϕ = ω(h+ vO(t, ϕ, h, ε)) +

3∑

j=2

∑

k,n∈Z

εjf jk,n(h) ei(kϕ+nt) + ε3 δ f3(t, ϕ, h, ε, δ) + ε4 f4(t, ϕ, h, ε)

˙h = ε2 g20,0(h) + ε2 δ2 g2(t, ϕ, h, ε, δ) + ε3 g30,0(h) + ε3 δ g3(t, ϕ, h, ε, δ) + ε4 g4(t, ϕ, h, ε)

(2.23)

defined for(

t, ϕ, h)

∈ UO :=

(

t, ϕ, h)

∈ R3∣

∣ h = h+ uO(t, ϕ, h, ε), h ∈ IO

ε,δ

. The maps gj0,0, gj are

bounded by a constant B∞(b∞, c1, c2, c3, c4).

2. Removing non–resonant terms on the ”inner region”

Let denote the constant given in lemma 2.2.2. Then there exists a constant εI > 0 such that forany |ε| ≤ εI and any resonance hm ∈ H the sets

I I

ε,δ = BR(hm, ), J I =

(k, n) ∈ Z∣

∣

∣

∣

(k, n) 6= (0, 0),−nk

6= ω(hm)

(2.24)

satisfy the assumptions of lemma 2.2.6 with p = 1, Iε,δ = I I

ε,δ and J = J I. The corresponding


transformed system takes the form

ϕ = ω(h+ vI(t, ϕ, h, ε)) +

3∑

j=2

∑

k,n∈Z

εjf jk,n(h) ei(kϕ+nt) + ε4f3(t, ϕ, h, ε) + ε4 f4(t, ϕ, h, ε)

˙h = ε2 g20,0(h) + ε2∑

l∈N∗

g2lkm,lnm(h) eil(kmϕ+nmt)

+ ε3 g30,0(h) + ε3∑

l∈N∗

g3lkm,lnm(h) eil(kmϕ+nmt)

+ ε4 g2(t, ϕ, h, ε) + ε4 g3(t, ϕ, h, ε) + ε4 g4(t, ϕ, h, ε),

(2.25)

where the integers nm < 0 < km have no common divisor and −nmkm

= ω(hm).

The function uI in the transformation h = h + uI(t, ϕ, h, ε) is of size O(ε2). More precisely thereexists a constant c6 > 0 such that the ”inverse mapping” vI of uI satisfies

∣

∣vI(t, ϕ, h, ε)∣

∣ ≤ ε2 c6 (2.26)

uniformly with respect to(

t, ϕ, h)

∈ U I :=

(

t, ϕ, h)

∈ R3∣

∣ h = h+ uI(t, ϕ, h, ε), h ∈ I I

ε,δ

. The

functions f jk,n, fj, gjk,n and gj are of class C 2 with respect to t, ϕ ∈ R and h ∈ BR(hm, ).

Moreover we recall the following statement proved in lemma 2.2.6 which is true in both situations:

if f2k,n = 0 for all (k, n) ∈ Z2 then f3 = 0 and similarly

if g2k,n = 0 for all (k, n) ∈ Jj, then g2 = 0.

Note however, that the functions f j, gj in the two situations listed above are not identical.

PROOF: Without loss of generality we may assume that ε 6= 0. (In the case were ε = 0 system (2.1)may be discussed directly). In a first step we consider the two cases listed in the statement separately:

1. p = 0, IO

ε,δ = R \M⋃

m=1BR(hm,

|ε|δ ) and J

O = Z \ (0, 0) :

Let , c1 ∈ (0, 1], c2 ≥ 1 and c4 denote the constants introduced in lemmata 2.2.1, 2.2.2 and 2.2.4

and define δ∞ := min

1,√

c13 c2 c4

, c1√8 c4 c2

, εO(δ) := min

1, 14 δ

as well as bO(ε, δ) := c2c1

δ|ε| . We

establish the assumptions made in lemma 2.2.6:

(a) The estimate (2.4) together with |ε|δ ≤ 1

4 < 1 yields

d(h) ≥ c1 min1≤m≤M

1, |h− hm| ≥ c1 min

1,|ε|δ

= c1|ε|δ

(2.27)

for every h ∈ IO

ε,δ. Since (0, 0) 6∈ JO and 1 < δ|ε| ≤ δ

|ε| ≤ δ|ε|

1c1

we find by lemma 2.2.2

1

|k ω(h) + n| ≤ c2 max

1,1

d(h)

≤ c2 max

1,δ

|ε| c1

=c2c1

δ

|ε| = bO(ε, δ).

This verifies assumption (2.14 a).


(b) Since |ε|δ <

14 δ

δ = 14 it is easy to see that the components of the set IO

ε,δ are disjoint. From

definition (2.11) of cIO(ε, δ) we find cIO(ε, δ) =2 |ε|3 δ . By consequence of the estimate

2 ε2 c4 bO(ε, δ) = 2 ε2 c4

c2c1

δ

|ε| = 2c2 c4c1

δ |ε| ≤ 2c2 c4c1

δ2∞|ε|δ

≤ 2 |ε|3 δ

= cIO(ε, δ)

as well as

2 ε2 c4 bO(ε, δ) (1 + bO(ε, δ)) = 2 ε2 c4

c2c1

δ

|ε|

(

1 +c2c1

δ

|ε|

)

≤ 2 |ε|3 δ

+ 2 c4

(

c2 δ

c1

)2

≤ 23

14 + 2 c4

(

c2c1

)2

δ2∞ ≤ 12

(2.14b) is proved at once.

2. p = 1, I I

ε,δ = BR(hm, ) and JI =

(k, n) ∈ Z | (k, n) 6= (0, 0), −nk 6= ω(hm)

:

In this case where p = 1, the parameter δ does not appear in the assumptions of lemma 2.2.6explicitely. The quantities f j , gj as in lemma 2.2.6 therefore do not depend on δ either.

(a) εI := min

1, 1√8 c4 c2

and bI(ε, δ) := c2. It then follows from (2.5 c) that 1|k ω(h)+n| ≤ bI(ε, δ)

for all h ∈ I I

ε,δ, (k, n) ∈ J I. This corresponds to assumption (2.14 a).

(b) Since I I

ε,δ consist of a single open interval, the first assumption made in (2.14 b) is trivial. Forthis choice of I I

ε,δ the integer L in lemma 2.2.5 equals 1 and hence cII = 1. Thus the estimates

2 ε2 c4 bI(ε, δ) = 2 ε2 c4 c2 ≤ 2 ε2 c4 (c2)

2 ≤ 14 < cII

and

2 ε2 c4 bI(ε, δ) (1 + bI(ε, δ)) ≤ 4 ε2 c4 (c2)

2 ≤ 12

prove (2.14b) at once.

In order to establish the last assumption (2.14 c) we first note that in both situations considered, b(ε, δ)may be represented as

b(ε, δ) = c2

(

δ

c1 |ε|

)1−p

implying

( |ε|δ

)2 (1−p)b(ε, δ) (1 + b(ε, δ)) =

( |ε|δ

)2 (1−p)c2

(

δ

c1 |ε|

)1−p+

( |ε|δ

)2 (1−p)(c2)

2

(

δ

c1 |ε|

)2 (1−p)

=

( |ε|δ

)1−pc2

(c1)1−p +

(

c2

(c1)1−p

)2

≤ c2

(c1)1−p +

(

c2

(c1)1−p

)2

≤ 2

(

c2c1

)2

and( |ε|δ

)1−pb(ε, δ) =

( |ε|δ

)1−pc2

(

δ

c1 |ε|

)1−p=

c2

(c1)1−p ≤ 2

(

c2c1

)2

.


Together with

δ

( |ε|δ

)p

≤ 1

assumption (2.14b) may be established by setting

c5 := max

1, 2

(

c2c1

)2

.

It remains to prove the explicit representations given for the transformed right–hand side. In the firstsituation it is easy to see that (JO)c = (0, 0) and by consequence of the statements given in lemma 2.2.6the transformed system takes the form (2.23). In the second situation we take into account that

(J I)c

= (k, n) ∈ Z | (k, n) 6∈ J I

=

(k, n) ∈ Z | (k, n) = (0, 0) or−nk

= ω(hm)

= (k, n) ∈ Z | k ω(hm) + n = 0 .

Hence (J I)cmay be represented in the form

(k, n) ∈ Z | ∃ l ∈ Z : (k, n) = l (km, nm)

where the integers nm < 0 < km have no common divisor. Applying lemma 2.2.6 we see that (2.16) takesthe form stated in (2.25).

In the second situation, the denominators arising in (2.15) are bounded uniformly by c2. Thus the mapuI is of class Cr. As r ≥ 3 the regularity of order C 2 claimed may be shown in a straightforward proof.We eventually recall the estimate (2.18) which in this case of bI(ε, δ) = c2 implies |u(t, ϕ, h, ε)| = O(ε2)at once. The corresponding claim on v follows by lemma 2.2.5.

Remark 2.2.8 Note that choosing any 0 < δ ≤ δ∞ and ε ∈ R such that |ε|δ < , the outer and inner

regions IO

ε,δ = R \M⋃

m=1BR(hm,

|ε|δ ) and I I

ε,δ = BR(hm, ) are overlapping. However, for the discussion of

system (2.25) it is sufficient to consider an appropriate O(ε)–neighbourhood of the resonance hm. More

precisely we will aim on the discussion of solutions of (2.25) with initial values h(t0) ∈ BR(hm, 2|ε|δ ).

Due to some technical reasons the discussion in section 2.3.2 starts with BR(hm, 4|ε|δ ), however. The

parameter δ first will be fixed in the discussion of the outer region (cf. section 2.3.1).

Systems of the form (2.1) considered here may be understood as systems with two angle coordinates (t, ϕ)where d

dt t = 1 and there exist only finitely many resonances. In this special case we have just shown, thata simple transformation reduces the discussion of the entire system to the analysis of the leading Fouriercoefficients of the vector field.

Following this way it is not necessary to calculate an approximation of the solutions, using the inner,outer and inner–outer asymptotic expansions and matching these expansions as proposed in many works(cf. e.g. [17]).


2.3 The Discussion of the Transformed Systems

Depending on the size and the sign of the Fourier–coefficient maps g20,0, g2lkm,lnm

in the systems (2.23)and (2.25) it is possible to draw conclusions on the asymptotic behaviour of solutions. These results thenmay be carried over to system (2.1).

2.3.1 The Behaviour in the Outer Regions

In this subsection we treat system (2.23) on the ”outer regions” which are at most O(ε)–close to aresonance. The following proposition considers the case of an existing ”minimal drift” on the set IO

ε,δ:

Proposition 2.3.1 Assume that there exists a constant c7 > 0 and that for Iε,δ ⊂ IO

ε,δ the estimate∣

∣g20,0(h)∣

∣ ≥ c7 holds for all h ∈ Iε,δ. Then δ > 0 may be choosen sufficiently small such that for

|ε| ≤ εO(δ), every solution of (2.23) with initial value in Iε,δ tends towards the border ∂Iε,δ.

More precisely if (ϕ, h)(t; t0, ϕ0, h0) denotes the solution of (2.23) with initial value (ϕ0, h0) at time t = t0where h0 ∈ Iε,δ then

∣

∣h(t; t0, ϕ0, h0)− h0∣

∣ ≥ ε2 12c7 (t− t0)

for all t ≥ t0 such that

h(s; t0, ϕ0, h0)∣

∣ s ∈ [t0, t]

⊂ Iε,δ.

PROOF: Let B∞ denote the uniform bound of the maps gj , g30,0 given by lemma 2.2.6. Since |ε| ≤min

1, 14 δ

the estimate

∣

∣ε2 δ2 g2(t, ϕ, h, ε, δ) + ε3 g30,0(h) + ε3 δ g3(t, ϕ, h, ε, δ) + ε4 g4(t, ϕ, h, ε)∣

∣

≤ ε2 δ2B∞ + ε3B∞ + ε3 δ B∞ + ε4B∞ ≤ ε2 δ B∞(

δ∞ + 14 + 1+ 1

4 )

≤ ε2 δ B∞ (δ∞ + 3)

hold. Hence for 0 < δ ≤ min

δ∞,c7

2B∞ (δ∞+3)

, ˙h is bounded from below :

∣

∣

∣

˙h∣

∣

∣ ≥∣

∣

∣

∣

∣ε2 g20,0(h)∣

∣−∣

∣ε2 δ g2(t, ϕ, h, ε, δ) + ε3 g30,0(h) + ε3 δ g3(t, ϕ, h, ε, δ) + ε4 g4(t, ϕ, h, ε)∣

∣

∣

∣

∣

≥ ε2 c7 − ε2 δ B∞ (δ∞ + 3) ≥ 12 ε

2 c7.

Thus for all t ≥ t0 such that (ϕ, h)(s; t0, ϕ0, h0) exists for all s ∈ [t0, t] we find ˙h(t; t0, ϕ0, h0) 6= 0 andtherefore

∣

∣h(t; t0, ϕ0, h0)− h0∣

∣ =

∣

∣

∣

∣

∣

∣

t∫

t0

˙h(s; t0, ϕ0, h0) ds

∣

∣

∣

∣

∣

∣

=

t∫

t0

∣

∣

∣

˙h(s; t0, ϕ0, h0)∣

∣

∣ ds

≥t∫

t0

12 ε

2 c7 ds = ε2 12c7 (t− t0).

2.3. The Discussion of the Transformed Systems 89

2.3.2 The Variables in the Inner Regions

Fixing any resonance hm ∈ H we now consider (2.25) in some neighbourhood of hm. More precisely we

perform a ”blow–up” of the resonance region BR(hm, 4|ε|δ ) of the h–variables. Replacing the variable ϕ

by the so–called resonance angle in the same step, this transforms (2.25) into a system where the leadingterms are of order O(ε) and autonomous. In these ”Inner Variables” it then will be possible to discussthe existence of solutions passing through the inner region. This will be carried out in section 2.3.3 andsection 2.3.4.

In this section as well as in section 2.3.3 and in section 2.3.4 the parameter δ is fixed according topropositions 2.2.7, 2.3.1 and such that some additional conditions are met. These conditions will bepointed out later and are independent of ε.

In order to make sure that the results obtained in proposition 2.2.7 may be applied we have to considervalues for ε such that in addition to |ε| ≤ εI the estimate

4|ε|δ

≤

applies.

Definition 2.3.2 Fixing any hm ∈ H, let km, nm denote the integers given by proposition 2.2.7. We

introduce the Inner Variables for system (2.25) in the inner region BR(hm, 4|ε|δ ) as follows:

ε h := kmddhω(hm)

(

h− hm)

∀∣

∣h− hm∣

∣ < 4|ε|δ

ψ := km ϕ+ nm t.

(2.28)

The angle ψ is usually refered to as the resonance angle.

Note that ddhω(hm) may be negative but does not vanish (cf. GA 2.5)

The following lemma gives an explicit representation of the O(ε) terms of the system corresponding to(2.25) in the new coordinates (ψ, h). This form may be sufficient for a qualitative discussion near theresonance hm, provided that at least one of the maps g20,0, g

2lkm,lnm

does not vanish in hm.

Lemma 2.3.3 Applying transformation (2.28), system (2.25) may be represented in the more conven-tional form

ψ = ε h+ ε2 f2(t, ψ, h, ε)

˙h = ε

(

a0 +∑

l∈N∗

acl cos(lψ) + asl sin(lψ)

)

+ ε2 g2(t, ψ, h, ε)(2.29)

defined for∣

∣

∣h∣

∣

∣ <4 |αm|δ where αm := km

ddhω(hm) and

a0 := g20,0(hm)αm

acl := 2ℜ(g2lkm,lnm(hm))αm asl := −2ℑ(g2lkm,lnm(hm))αm.(2.30)

The maps f , g are of class BC 2 for t, ψ ∈ R, h ∈ 4 |αm|δ and km 2π–periodic with respect to t and ψ.


PROOF: Let (ϕ, h)(t) be a solution of (2.25) and consider (ψ, h)(t) as defined by (2.28). Applying theidentity km ω(hm) + nm = 0 we then find

ψ = km(

ω(h+ vI(t, ϕ, h, ε))− ω(hm))

+ km ω(hm) + nm

+ km

3∑

j=2

∑

k,n∈Z

εjf jk,n(h) ei(kϕ+nt) + ε4 km f

3(t, ϕ, h, ε) + ε4 km f4(t, ϕ, h, ε)

= kmddhω(hm)

(

h+ vI(t, ϕ, h, ε)− hm)

+

1∫

0

(1 − σ) d2

dh2ω(

hm + σ(

h+ vI(t, ϕ, h, ε)− hm))

dσ(

h+ vI(t, ϕ, h, ε)− hm)2

+ km

3∑

j=2

∑

k,n∈Z


3(t, ϕ, h, ε) + ε4 km f4(t, ϕ, h, ε).

Hence with h+ vI(t, ϕ, h, ε)− hm = ε h/(

kmddhω(hm)

)

+ vI(t, ϕ, h, ε) we find

ψ = ε h+ kmddhω(hm) vI(t, ϕ, h, ε)

+

1∫

0

(1 − σ) d2

dh2ω(

hm + σ(

h+ vI(t, ϕ, h, ε)− hm))

dσ(

ε h/(

kmddhω(hm)

)

+ vI(t, ϕ, h, ε))2

+km

3∑

j=2

∑

k,n∈Z


3(t, ϕ, h, ε) + ε4 km f4(t, ϕ, h, ε).

We proceed in a similar way to obtain the result claimed for the h equation:

˙h

kmddhω(hm)

=1

ε˙h = ε g20,0(hm) + ε

1∫

0

ddhg

20,0

(

hm + σ(

h− hm))

dσ(

h− hm)

+ε∑

l∈N∗

g2lkm,lnm(hm)eilψ

+ε∑

l∈N∗

[

1∫

0

ddhg

2lkm,lnm

(

hm + σ(

h− hm))

dσ(

h− hm) ]

eilψ

+ε2 g30,0(h) + ε2∑

l∈N∗

g3lkm,lnm(h)eilψ + ε3 g2(t, ϕ, h, ε) + ε3 g3(t, ϕ, h, ε)

+ε3 g4(t, ϕ, h, ε).

Recall that the right hand side of the equations for ψ and ˙h is 2π–periodic with respect to t and ϕ. Hencereplacing the argument ϕ by (ψ − nm t) /km (cf. (2.28)) the resulting expressions are km 2π–periodic withrespect to t and ψ. Moreover as we have proved in (2.26),

∣

∣vI(t, ϕ, h, ε)∣

∣ ≤ ε2 c6 and therefore we are ableto rewrite (2.25) in the form

ψ = ε h+ ε2 f2(t, ψ, h, ε)

˙h = ε

(

g20,0(hm) +∑

l∈N∗

g2lkm,lnm(hm)eilψ

)

(

kmddhω(hm)

)

+ ε2 g2(t, ψ, h, ε).(2.31)


In view of the uniform boundedness of the maps ddhg

jk,n and taking into account that

∣

∣

∣h∣

∣

∣ ≤ 4 |αm|δ is a

compact domain of h, it then follows at once that the maps f , g are bounded uniformly.

As the right hand side of (2.1) has to be real, it follows that g2lkm,lnm(hm) = g2−lkm,−lnm(hm) (i.e. thecomplex conjugate value). Then the representation as sine / cosine–series given in (2.29) follows at once.

In the discussion to follow we will consider the case where

acl = asl = 0 for l ≥ 2. (2.32)

This will be sufficient to apply the results obtained here in many situations, as for instance in theexample of a synchronous motor, presented in chapter 4. However, the reader will be able to dealwith the more general case by carrying over the process given here.

Note that in this case of (2.32) the equations (2.29) found in a neighbourhood of the resonances are ofperturbed ”pendulum type”: the quantity a0 corresponds to ”the torque” of the pendulum and ac1, a

s1

are defined by the ”acceleration of gravity”.


2.3.3 The Case of Complete Passage through the Inner Regions

The aim of this section is to show that in the case where |a0| <√

(ac1)2+ (as1)

2, all solutions (up to a set

of size O(ε)) pass through the inner region. For simplicity we consider the case where (2.32) holds.

Lemma 2.3.4 Assume that in a resonance hm ∈ H the ”mean value” dominates the ”resonant terms”,i.e.

|a0| >√

(ac1)2 + (as1)

2. (2.33)

Then the perturbation parameter ε may be choosen sufficiently small such that all solutions of (2.29) passthe resonance region within finite time (of size O(1/ε2)).

Note that the statement of lemma 2.3.4 together with the definition of the coordinate h in (2.28) implies

that the solutions of (2.25) starting in BR(hm, 2|ε|δ ) pass the resonance region BR(hm, 2

|ε|δ ).

PROOF: By assumption we have |a0| >√

(ac1)2+ (as1)

2and hence |a0| > |ac1 cos(ψ) + as1 sin(ψ)|. The

constant Fourier term a0 thus dominates the remaining terms in (2.29) if ε is choosen small and the proofmay be carried out in a very similar way as given in the proof of proposition 2.3.1.

The following figure illustrates the phase portrait of (2.29) when omitting O(ε2)–terms in this situationof passage through the inner region.

Figure 2.3: |a0| >√

(ac1)2+ (as1)

2: no fixed points on resonance h = 0 hence passage through the inner

region for all solutions.


2.3.4 The Case of Passage for all Solutions up to a O(ε)–Set

For simplicity we consider the case where (2.32) holds.

We continue the qualitative discussion of system (2.29) by considering the case where the resonant termsdominate the mean value a0. Without loss of generality we assume that

sgn(a0) = −1.

The following plot illustrates the phase portrait of the O(ε)–terms of (2.29) in this case.

Figure 2.4: a0 < 0 and |a0| <√

(ac1)2+ (as1)

2: hyperbolic and elliptic fixed point on resonance h = 0.

It may be seen that omitting O(ε2) terms in (2.29) there exist hyperbolic equilibria at (ψ∗ + j 2π, 0),j ∈ Z. Hence the complete system (2.29) admits a collection of hyperbolic, km 2π–periodic solutionsnear (ψ∗ + j 2π, 0). In a next step we will apply a time dependent translation on (2.29) such that theseperiodic solutions become equilibria again. This is the subject of the following lemma:


Lemma 2.3.5 Consider system (2.29) and assume that |a0| <√

(ac1)2 + (as1)

2. Then there exists ψ∗ ∈[0, 2π), ε1 > 0 and a map Φ ∈ C 2(R×R× (−ε1, ε1),R2) satisfying Φ(t, ψ, 0) = 0 which is km 2π–periodicwith respect to t and ψ such that the following statement is true:

The equation

ξ = (ξ1, ξ2) = (ψ, h)− Φ(t, ψ, ε) (2.34)

defines a change of coordinates for ψ ∈ R,∣

∣

∣h∣

∣

∣ <4 |αm|δ . It transforms system (2.29) to a system of the

form

ξ = ε J∇H(ξ) + ε2 ∆(t, ξ, ε) (2.35)

defined for ξ ∈ R× [−αδ,m, αδ,m] where αδ,m := 3 |αm|δ and H(ξ) := 1

2 ξ22 −(a0 ξ1+a

c1 sin(ξ1)−as1 cos(ξ1)).

In particular there exists a set

ξjH

∣

∣

∣ j ∈ Z

of hyperbolic fixed points ξjH = (ψ∗ + j 2π, 0) of (2.35). The

map ∆ is of class BC 1 and km 2π–periodic with respect to t and ξ1.

Finally, the transformation defined via (2.34) maps the region∣

∣

∣h∣

∣

∣ <2 |αm|δ into the ”strip” |ξ2| < αδ,m.

PROOF: We content ourselves here with a sketch of the proof.

1. Existence of ψ∗ : It is elementary that by consequence of |a0| <√

(ac1)2+ (as1)

2there exists a zero

of the function ψ 7→ a0 + ac1 cos(ψ) + as1 sin(ψ) such that the derivative in ψ∗ is positive, i.e.

−ac1 sin(ψ∗) + as1 cos(ψ∗) > 0. (2.36)

2. Existence of periodic solutions of (2.29): Fixing any 0 ≤ j ≤ km − 1 we set x = (x1, x2) =(ψ − ψ∗ − j 2π, h),

f(x) :=

(

x2a0 + ac1 cos(ψ∗ + x1) + as1 sin(ψ∗ + x2)

)

g(x, t, ε) :=

(

ε f2(t, ψ∗ + x1, x2, ε)ε g2(t, ψ∗ + x1, x2, ε)

)

.

Then f(0) = 0, detDf(0) = ac1 sin(ψ∗)− as1 cos(ψ∗) 6= 0 and (2.29) is equivalent to

x = ε (f(x) + g(x, t, ε)) . (2.37)

Together with the properties of f2, g2 proved in lemma 2.3.3, the assumptions of lemma 1.2.1 maybe verified for (2.37) with p = 1, r = 2 and T = km 2π. Hence it is a consequence of lemma 1.2.1that for |ε| sufficiently small, there exists a km 2π–periodic solution xj(t, ε) of system (2.37) of classC2, satisfying xj(t, 0) = 0.

As we chose 0 ≤ j ≤ km − 1 arbitrary and the vector field in (2.29) is km 2π–periodic with respectto ψ it follows that there exists a family

(

ψj , hj) ∣

∣ j ∈ Z

of km 2π–periodic solutions of (2.29)

where(

ψj+km , hj+km)

(t, ε)−(

ψj , hj)

(t, ε) = (km 2π, 0) and(

ψj , hj)

(t, 0) = (ψ∗ + j 2π, 0).


3. Existence of the transformation Φ: Consider a function χ ∈ BC 2(R,R) with the following properties

• χ(s) = 1 if |s| ≤ 2π/3

• χ(s) = 0 if |s| ≥ 4 π/3

• χ(s) + χ(s− 2π) = 1 for all s ∈ [0, 2π] 2π3

4π3

2π3

4π3

s

2π0--

and define

Φ(t, ψ, ε) :=∑

j∈Z

χ(ψ − ψ∗ − j 2π)[ (

ψj , hj)

(t, ε)− (ψ∗ + j 2π, 0)]

.

Note that this definition implies

Φ(t, ψj(t, ε), ε) =(

ψj , hj)

(t, ε)− (ψ∗ + j 2π, 0) ∀j ∈ Z. (2.38)

It then is straightforward to establish the statement claimed on the existence of the transformation(2.34) and the form of the transformed vector field, as presented in (2.35).

4. Hyperbolicity of the fixed points: By consequence of (2.38) we find (ψ, h)(t) =(

ψj , hj)

(t, ε) to beequivalent to

ξ(t) =(

ψj , hj)

(t, ε)− Φ(t, ψj(t, ε), ε) = (ψ∗ + j 2π, 0) ∀t ∈ R

and thus ξjH := (ψ∗+ j 2π, 0) to be a fixed points of (2.35). Expanding the characteristic exponents

of the linearization of (2.35) at ξ = ξjH with respect to ε yields

±ε√

−ac1 sin(ψ∗ + j 2π) + as1 cos(ψ∗ + j 2π) +O(ε2).

Thus by (2.36) there exists ε1 > 0 such that for 0 6= |ε| < ε1 these eigenvalues have non–zero realvalues and are of opposite sign.

Recall that by lemma 2.3.3, system (2.29) is defined for∣

∣

∣h∣

∣

∣ ≤ 4 |αm|δ . Hence we may choose ε1 sufficiently

small such that the images of the region∣

∣

∣h∣

∣

∣ <2 |αm|δ applying the transformation (2.34) are contained in

the ”strip” |ξ2| < αδ,m = 3 |αm|δ . Without loss of generality we furthermore may assume that the image

of∣

∣

∣h∣

∣

∣≤ 4 |αm|

δ contains the set 3 |αm|δ such that the map ∆(t, ξ, ε) is defined for |ξ2| ≤ αδ,m. This proves

the statements given in lemma 2.3.5.

Note that by consequence of lemma 2.3.5 every solution of (2.25) with initial value∣

∣h− hm∣

∣ < 2 |ε|δ

corresponds to a unique solution of (2.35) with initial value |ξ2| < αδ,m. This makes it possible to obtainqualitative results on (2.25) by discussing system (2.35).

Definition 2.3.6 For technical reason we define regions Cjδ , j ∈ Z together with their upper bound-

aries Ajδ in the strip |ξ2| ≤ αδ,m using the (un)stable manifolds of the hyperbolic fixed points ξjH of the

”unperturbed” autonomous Hamiltonian system

ξ = ε J∇H(ξ)

as illustrated in the following figure:


δ

Cj

δ

α

H H

δA

ξj+1ξ

j

j

ξ1

,m

Figure 2.5: Definition of the sets Ajδ and Cjδ , illustrated in the case of a0 < 0.

(Without any loss of generality we assume that in the beginning of section 2.3.2 the quantity δ has beenchosen sufficiently small such that the strip |ξ2| ≤ αδ,m covers the homoclinic orbits and the situation isas depicted here, indeed.)

Lemma 2.3.7 There exists a constant ∆ > 0 such that

|∆(t, ξ, ε)| ≤ ∆∣

∣

∣∇H(ξ)∣

∣

∣

holds for ξ ∈ ⋃

j∈Z

Cjδ .

PROOF: Note first that by definition of H, the map ξ 7→ D3H(ξ) does not depend on ξ2 explicitely andis 2π–periodic with respect to ξ1. Hence

sup

∣

∣

∣D3H(ξ)∣

∣

∣

∣

∣

∣

∣

∣

∣

ξ ∈⋃

j∈Z

Cjδ

≤ supξ∈R2

∣

∣

∣D3H(ξ)∣

∣

∣ = b1 <∞

where b1 > 0. As the Hessian matrix D2H(ξjH) =

[

ac1 sin(ψ∗)− as1 cos(ψ∗) 00 1

]

is regular (cf. (2.36)) we

find b2 := 1/(

2 b1

∣

∣

∣D2H(ξjH)−1∣

∣

∣

)

<∞, independent of j. By definition of the sets Cjδ ,

∇H(ξ) = 0, ξ ∈⋃

j∈Z

Cjδ ⇐⇒ ξ ∈⋃

j∈Z

ξjH. (2.39)

Thus b3 := inf

∣

∣

∣∇H(ξ)∣

∣

∣

∣

∣

∣

∣

∣

ξ ∈ ⋃

j∈Z

Cjδ ,∣

∣

∣ξ − ξjH

∣

∣

∣ ≥ b2 ∀j ∈ Z

is positive. We consider two cases:

1. Let ξ ∈ ⋃

j∈Z

Cjδ with∣

∣

∣ξ − ξjH

∣

∣

∣ ≥ b2 for all j ∈ Z. Since ∆ is periodic in t, ψ we have

|∆(t, ξ, ε)| ≤ b4 := sup

|∆(t, ξ, ε)|

∣

∣

∣

∣

∣

∣

t ∈ R, ξ ∈⋃

j∈Z

Cjδ , |ε| ≤ ε1

<∞,


hence by definition of b3 |∆(t, ξ, ε)| ≤ b4b3

∣

∣

∣∇H(ξ)∣

∣

∣.

2. On the other hand, if for j ∈ Z fixed, ξ ∈ ⋃

j∈Z

Cjδ satisfies∣

∣

∣ξ − ξjH

∣

∣

∣ ≤ b2, then

b5 := sup

|∂ξ∆(t, ξ, ε)|

∣

∣

∣

∣

∣

∣

t ∈ R, ξ ∈⋃

j∈Z

BR2(ξjH , b2), |ε| ≤ ε1

<∞.

Since ∇H(ξjH) = 0, for every ξ ∈ ⋃

j∈Z

Cjδ there exists an ξ(ξ, ξjH) such that

∣

∣

∣∇H(ξ)

∣

∣

∣=

∣

∣

∣

∣

D2H(ξjH)(

ξ − ξjH

)

+D3H(ξ(ξ, ξjH))(

ξ − ξjH

)[2]∣

∣

∣

∣

≥∣

∣

∣

∣

∣

∣

∣D2H(ξjH)(

ξ − ξjH

)∣

∣

∣ −∣

∣

∣

∣

D3H(ξ(ξ, ξjH))(

ξ − ξjH

)[2]∣

∣

∣

∣

∣

∣

∣

∣

≥(

1/∣

∣

∣D2H(ξjH)−1

∣

∣

∣− b1

∣

∣

∣ξ − ξjH

∣

∣

∣

) ∣

∣

∣ξ − ξjH

∣

∣

∣

≥(

1/∣

∣

∣D2H(ξjH)−1∣

∣

∣− b1 b2

) ∣

∣

∣ξ − ξjH

∣

∣

∣

=∣

∣

∣ξ − ξjH

∣

∣

∣ /(

2∣

∣

∣D2H(ξjH)−1∣

∣

∣

)

.

Using ∆(t, ξjH , ε) = 0 (lemma 2.3.5) we obtain the inequality

|∆(t, ξ, ε)| ≤ b5

∣

∣

∣ξ − ξjH

∣

∣

∣ ≤ b5 2∣

∣

∣D2H(ξjH)−1∣

∣

∣

∣

∣

∣∇H(ξ)∣

∣

∣ .

Setting ∆ := max

b4b3, 2 b5

∣

∣

∣D2H(ξjH)−1∣

∣

∣

the claim is established.

Lemma 2.3.8 Let H denote the Hamiltonian introduced in lemma 2.3.5 and ∆ be the constant given bylemma 2.3.7. Setting

w(ξ, ε) :=∆√

1− ε2 ∆2∇H(ξ) for 0 < ε < 1/∆, (2.40)

the following statements hold for all1 t ∈ R, ξ ∈ ⋃

j∈Z

Cjδ , 0 < ε ≤ ε2 := min

ε1,1

2 ∆

and any fixed Λ > 0

2.41 a.(

ε J∇H(ξ) + ε2∆(t, ξ, ε))

∧(

ε J∇H(ξ) + ε2 (1 + Λ) w(ξ, ε))

≥ ε3 12 Λ ∆

∣

∣

∣∇H(ξ)

∣

∣

∣

2

2.41 b.(

ε J∇H(ξ) + ε2∆(t, ξ, ε))

∧(

ε J∇H(ξ)− ε2 (1 + Λ) w(ξ, ε))

≤ −ε3 12 Λ ∆

∣

∣

∣∇H(ξ)∣

∣

∣

2

.

PROOF: It easily may be found that for any fixed ξ ∈ ⋃

j∈Z

Cjδ , the vectors ε J∇H(ξ)+ε2 w(ξ, ε) correspond

to the tangents as illustrated in the figure 2.6:

1For simplicity we will consider non–negative ε in what follows. The procedure given here may be carried over to thecase of ε < 0 by appropriately adapting the signs during the discussion.


ξ εε2

∆ ξw( , )

H( )+Jε

∆ ξH( )Jε

ε 2

∆

ξξ εw( , )

H( )-Jε

ε2 ξ εw( , )∆

ξε 2

H( )

∆

∆

ξH( )+ε ε2∆ (t, , )ξ εJ

Figure 2.6: Illustration of the tangents ε J∇H(ξ) ± ε2 w(ξ, ε) on the circle of radius ε2 ∆∣

∣

∣∇H(ξ)

∣

∣

∣

centered at ε J∇H(ξ).

By definition of w(ξ, ε) we then see that

∣

∣

∣ε J∇H(ξ) + ε2 w(ξ, ε)∣

∣

∣ =

√

∣

∣

∣εJ∇H(ξ)∣

∣

∣

2

+ |ε2 w(ξ, ε)|2 =ε√

1− ε2 ∆2

∣

∣

∣∇H(ξ)∣

∣

∣ . (2.42)

Let us establish the first statement (2.41 a):(

ε J∇H(ξ) + ε2 ∆(t, ξ, ε))

∧(

ε J∇H(ξ) + ε2 (1 + Λ) w(ξ, ε))

= ε J∇H(ξ) ∧ ε J∇H(ξ) + ε J∇H(ξ) ∧ ε2 (1 + Λ) w(ξ, ε)

+ ε2∆(t, ξ, ε) ∧(

ε J∇H(ξ) + ε2 w(ξ, ε))

+ ε2 ∆(t, ξ, ε) ∧ ε2Λw(ξ, ε)

= ε J∇H(ξ) ∧ ε2 (1 + Λ) ∆/√

1− ε2 ∆2 ∇H(ξ)

+ ε2∆(t, ξ, ε) ∧(

ε J∇H(ξ) + ε2 w(ξ, ε))

+ ε2 ∆(t, ξ, ε) ∧ ε2Λw(ξ, ε)

≥ ε3 (1 + Λ) ∆/√

1− ε2 ∆2∣

∣

∣∇H(ξ)∣

∣

∣

2

−∣

∣ε2 ∆(t, ξ, ε)∣

∣

∣

∣

∣ε J∇H(ξ) + ε2 w(ξ, ε)

∣

∣

∣−∣

∣ε2∆(t, ξ, ε) ∧ ε2 Λw(ξ, ε)∣

∣ .

Considering points ξ ∈ ⋃

j∈Z

Cjδ we may use the estimate proved in lemma 2.3.7. This together with (2.42)

leads to(

ε J∇H(ξ) + ε2∆(t, ξ, ε))

∧(

ε J∇H(ξ) + ε2 (1 + Λ) w(ξ, ε))

≥ ε3 (1 + Λ)∆√

1− ε2 ∆2

∣

∣

∣∇H(ξ)∣

∣

∣

2

− ε2 ∆∣

∣

∣∇H(ξ)∣

∣

∣

ε√1− ε2 ∆2

∣

∣

∣∇H(ξ)∣

∣

∣

−ε4 Λ ∆∣

∣

∣∇H(ξ)∣

∣

∣

∆√1− ε2 ∆2

∣

∣

∣∇H(ξ)∣

∣

∣

= ε3 Λ∆√

1− ε2 ∆2

(

1− ε ∆)

∣

∣

∣∇H(ξ)∣

∣

∣

2

≥ ε3 12 Λ ∆

∣

∣

∣∇H(ξ)∣

∣

∣

2

.

The second statement (2.41b) is proved in an analogous way.


We continue with our task by considering the autonomous systems

2.43 a. ξ = ε J∇H(ξ) + ε2 (1 + Λ) w(ξ, ε)

2.43 b. ξ = ε J∇H(ξ)− ε2 (1 + Λ) w(ξ, ε)

It may easily be checked that ξjH , j ∈ Z are hyperbolic fixed points of (2.43 a), (2.43 b) respectively. Thus

there exist the stable manifold Uj+1,++,ε and the unstable manifold Uj+1,−

+,ε of ξj+1H for system (2.43 a) as

well as the stable manifold Uj,+−,ε and the unstable manifold Uj,−−,ε of ξjH for system (2.43 b). With the help

of these manifolds we now define the strip of passage Djε,δ, bounded by Uj,±−,ε and Uj+1,±

+,ε as illustrated infigure 2.7.

Q ( )Q ( )

A

P( )1 0 P( )1 ε

H

ε0

−,ε

δ

ξ

1

j

1

j

j,-

H

j+1

U

ξ

jD

α δ,m

ξ1

ε,δU

j,+−,ε

Uj+1,-

j+1,+U +,ε

+,ε

Figure 2.7: Definition of the set Djε,δ using the invariant manifolds of ξjH , ξj+1

H with respect to systems

(2.43 a), (2.43 b) respectively (in the case a0 < 0). The curves plotted light grey depict level curves of H.

By consequence of lemma 2.3.8 the vector field ε J∇H(ξ) + ε2 ∆(t, ξ, ε) of system (2.35) evaluated onthe boundaries Uj+1,±

+,ε and Uj,±−,ε of Djε,δ points strictly into Dj

ε,δ for all t ∈ R. Hence Djε,δ is positively

invariant with respect to (2.35). This is proved via the following general result.

Lemma 2.3.9 Assume that we are given constants δ0, δ1 > 0, maps F ∈ C1(Q,R2) and G ∈ C1(R ×Q,R2) (where Q := (−δ1, δ1) × (−δ1, δ1)) as well as s+0 , s

−0 ∈ C1((−δ1, δ1), (−δ1, δ1)) together with a

family sα ⊂ C1((−δ1, δ1), (−δ1, δ1)), α ∈ (0, δ0] satisfying sα(0) = α such that the following statementsare true.


1. The system

ξ = F (ξ), ξ ∈ Q (2.44)

admits a hyperbolic fixed point at ξ = 0. The stable manifold U+0 and the unstable manifold U−

0 ofξ = 0 are identical to graph(s+0 ), graph(s

−0 ), respectively. The graph Uα := graph(sα) is invariant

with respect to (2.44) for every 0 < α ≤ δ0.

2. The system

ξ = F (ξ) +G(t, ξ), ξ ∈ Q (2.45)

admits a hyperbolic fixed point at ξ = 0. We denote the corresponding local stable manifold at timet0 by U+

t0 .

3. Let2 U+0,r := U+

0 ∩R+ ×R, U−0,l := U−

0 ∩R− ×R and S := U+0,r ∪ U−

0,l ∪⋃

α∈(0,δ0]

Uα. For every t ∈ R,

ξ ∈ S \ 0 the inequality

F (ξ) ∧G(t, ξ) < 0 (2.46)

applies. Finally the first component of the vector F (ξ) is not positive for all ξ ∈ S.

Then the local stable manifold of the hyperbolic fixed point ξ = 0 of system (2.45) lies outside the set Sfor every t0, i.e.

U+t0 ∩ S = 0

for all t0 in R.

1

δ1

U+t

δ

δ0

U

0

U+

^

α

ξ 1

ξ 2

F( )+G(t, )ξ ξ

ξ0

U+0 U

,r

0

F( )

−0,l

−

U

α

Q

S

Figure 2.8: Illustration of the situation considered in lemma 2.3.9.

2where R−

:= t ∈ R | t ≤ 0


PROOF: The proof of this lemma is carried out in two steps. We first establish that for every 0 < α ≤ δ0the set

Mα := (t, ξ1, ξ2) ∈ R×Q | ξ2 ≥ sα(ξ1)

is positive invariant with respect to the system (2.45) (written in autonomous form). This is shown byapplying theorem (16.9) in [1]. In a first step we fix any 0 < α ≤ δ0 and define the set X := R × Q aswell as the map

Φ(t, ξ) := sα(ξ1)− ξ2

for (t, ξ) ∈ X . It then is evident that Mα = Φ−1((−∞, 0]) and Φ ∈ C1(X,R). As the gradient of Φ isgiven by

∇Φ(t, ξ) =

0∂ξ1sα(ξ1)

−1

it does not vanish on X . Since Φ−1(0) = R×Uα and the vector F (ξ1, sα(ξ1)) is tangent to Uα for every|ξ1| < δ1 we have

(

∇Φ(t, ξ)

∣

∣

∣

∣

(

0F (ξ)

))

= 0 ∀(t, ξ) ∈ R× Uα.

Taking into account that the first component of F (ξ) is not positive on Uα we conclude

|F (ξ)| ∇Φ(t, ξ) = |∇Φ(t, ξ)|(

0−JF (ξ)

)

∀(t, ξ) ∈ R× Uα.

Hence we find by (2.46)

|F (ξ)|(

∇Φ(t, ξ)

∣

∣

∣

∣

(

1F (ξ) +G(t, ξ)

))

= |∇Φ(t, ξ)|((

0−JF (ξ)

) ∣

∣

∣

∣

(

1F (ξ) +G(t, ξ)

))

= |∇Φ(t, ξ)| F (ξ) ∧G(t, ξ) < 0

for all (t, ξ) ∈ R× Uα = ∂Mα. Hence theorem (16.9) in [1] may be applied here and implies the positiveinvariance of Mα with respect to the autonomous system

d

dt(t, ξ) = f(t, ξ) = (1, F (ξ) +G(t, ξ)) (2.47)

defined on X . Since α ∈ (0, δ0] was chosen arbitrary this is true for all α ∈ (0, δ0].

In a second step we prove the claim made in lemma 2.3.9 by contradiction. Assume that there existst0 ∈ R and ξ0 ∈ U+

t0 ∩ S with ξ0 6= 0. If on one hand ξ0 ∈ U+0,r ∪ U−

0,l then

F (ξ0) ∧(

F (ξ0) +G(t0, ξ0))

= F (ξ0) ∧G(t0, ξ0) < 0

i.e. the tangent vector at (t0, ξ0) on the orbit of the solution ξ(t; t0, ξ0) with initial value (t0, ξ0) of (2.47)points strictly into the set

M0 :=

(t, ξ1, ξ2) ∈ X∣

∣ξ2 ≥ max

s+0 (ξ1), s−0 (ξ1)

.


Hence there exist t1 > t0, α ∈ (0, δ0] such that ξ1 := ξ(t1; t0, ξ0) ∈ Mα. However, as ξ0 ∈ U+t0 we find

ξ1 ∈ U+t1 ∩Mα. If on the other hand ξ0 6∈ U+

0,r ∪ U−0,l then ξ

0 ∈ ⋃

α∈(0,δ0]

Uα. Hence there exists α ∈ (0, δ0]

such that ξ1 := ξ0 is an element of U+t1 ∩Mα at time t1 := t0.

For every point ξ1 ∈ U+t1 ∩Mα the limit of ξ(t; t1, ξ1) for t→ ∞ exists and corresponds to the origin. As

α 6= 0 and sα is continuous the set Mα is bounded away from the origin and thus the solution ξ(t; t1, ξ1)leaves Mα. This contradicts to the positive invariance of Mα proved in the first step.

We apply this result in order to prove that Djε,δ is positively invariant with respect to (2.35).

Lemma 2.3.10 The set Djε,δ is positively invariant with respect to system (2.35). Moreover there exists

ε3 > 0 such that if 0 < ε < ε3, all solutions ξ(t) of (2.35) starting in Ajδ ∩ Dj

ε,δ eventually cross the setξ2 = −αδ,m i.e. pass the resonance region.

PROOF: Let us first introduce the abbreviations

F (ξ, ε) := ε J∇H(ξ) + ε2 (1 + Λ) w(ξ, ε) and G(t, ξ, ε) := ε2 ∆(t, ξ, ε)− ε2 (1 + Λ) w(ξ, ε).

Calculating(

ε J∇H(ξ) + ε2 ∆(t, ξ, ε))

∧(

ε J∇H(ξ) + ε2 (1 + Λ) w(ξ, ε))

= −[

− ε J∇H(ξ) ∧ ε2 (1 + Λ) w(ξ, ε) + ε J∇H(ξ) ∧ ε2∆(t, ξ, ε) + ε2 (1 + Λ) w(ξ, ε) ∧ ε2 ∆(t, ξ, ε)]

= −[(

ε J∇H(ξ) + ε2 (1 + Λ) w(ξ, ε))

∧(

ε2 ∆(t, ξ, ε)− ε2 (1 + Λ) w(ξ, ε))

]

= −F (ξ, ε) ∧G(t, ξ, ε)

it follows from lemma 2.3.8 that

F (ξ, ε) ∧G(t, ξ, ε)

< 0 ξ ∈ Cjδ \ ξjH , ξ

j+1H

= 0 ξ ∈ ξjH , ξj+1H . (2.48)

Using the abbreviations F , G we rewrite system (2.35) in autonomous form:

t = 1, ξ = F (ξ, ε) +G(t, ξ, ε). (2.49)

An outer orthogonal vector to the manifold R × Uj+1,++,ε in (t, ξ) is given by

(

0−JF (ξ, ε)

)

whereas the

vector

(

1F (ξ, ε) +G(t, ξ, ε)

)

is tangent to the trajectory of (2.49) through (t, ξ) ∈ R× Uj+1,++,ε . Since

((

0−JF (ξ, ε)

) ∣

∣

∣

∣

(

1F (ξ, ε) +G(t, ξ, ε)

))

= F (ξ, ε) ∧G(t, ξ, ε) ≤ 0 ∀ ξ ∈ Uj+1,++,ε

this implies that the vector field of (2.49) does not point outside R×Uj+1,++,ε . Using analogous arguments

on Uj+1,−+,ε , Uj,+−,ε and Uj,−−,ε it follows that Dj

ε,δ is positively invariant with respect to system (2.49), (2.35)respectively.


Q ( )Q ( )

m

ε1

−,εj,-

U

α

α

m

ξ

δ,

j

U

P( )

jT

δ,

0

ξH

1

H

0

^ j,+U ε,

t

j+1,-

^ j+1,+U

1

j

+,ε

ε,3

ε,2Tε,4

jTε,5

j+1

C

P( )

j

jTε,1

0

ε1

t

0

A

j

δ

ε,T

j

δ

ξ1

Uj,+

−,εj+1,+

U +,ε

β

β

α

Figure 2.9: Illustration of the sets T jε,k ⊂ Dj

ε,δ considered in the proof of lemma 2.3.10.

The hyperbolic fixed point solution ξ = ξj+1H of (2.35) admits a time–dependent local stable manifold

(provided that 0 < ε < ε3 and ε3 is chosen sufficiently small). We denote its intersection with t = t0by Uj+1,+

ε,t0 . In a similar way there exists the intersection Uj,+ε,t0 of the stable manifold of ξjH with t = t0.

In view of (2.48) it is possible to transform system (2.35) in a neighbourhood of ξj+1H such that all

assumptions of lemma 2.3.9 are fulfilled. As a very similar transformation will be carried out explicitelybelow we refer the reader to the proof of proposition 2.3.11. By consequence of lemma 2.3.9 the setsUj+1,++,ε and Uj+1,+

ε,t0 at any time t0 are locally situated as depicted in figure 2.9. Using the same arguments

it is possible to establish the arrangement of the curve Uj,+ε,t0 of ξjH as depicted in figure 2.9. In particular

we find the curves Uj,+ε,t0 and Uj+1,+ε,t0 to intersect the positively invariant set Dj

ε,δ in ξjH , ξj+1H solely.

Let α ∈ (ξjH,1, ξj+1H,1 ) be the unique number such that ∇H(α, 0) = 0 (i.e. (α, 0) corresponds to the elliptic

fixed point of ξ = ε J∇H(ξ) situated between ξjH and ξj+1H ). In addition there exists a constant β > 0

such that the regions

T jε,1 := Dj

ε,δ ∩ [ξj+1H,1 − β, ξj+1

H,1 )× R and T jε,2 := Dj

ε,δ ∩ (−∞, α)× [−β, 0)

are contained in the neighbourhoods where lemma 2.3.9 is applied and that the qualitative behaviourinside T j

ε,1, T jε,2 is determined by the linearization of (2.35) in ξjH , ξj+1

H respectively. It then may be seen

that due to the position of the curves Uj,+ε,t0 , Uj+1,+ε,t0 every solution of (2.35) with initial value ξ0 ∈ T j

ε,1

(ξ0 ∈ T jε,2 respectively) leaves T j

ε,1 (T jε,2) within finite time.

We show that the same statement is true for the sets

T jε,3 := Dj

ε,δ ∩ (−∞, α]× R+ T jε,4 := Dj

ε,δ ∩ [α, ξj+1H,1 − β]× R T j

ε,5 := Djε,δ ∩ (−∞, α]× (−∞,−β].


We start with T jε,3. It is easy to see that there exists a constant 0 < b0 < ∞ such that for every

ξ ∈ Cjδ,3 := Cjδ ∩ (−∞, α]× R+ the estimate∣

∣

∣ξ − ξjH

∣

∣

∣≤ b0 ξ2 holds. Hence from lemma 2.3.7

|∆(t, ξ, ε)| ≤ ∆∣

∣

∣∇H(ξ)∣

∣

∣ ≤ ∆ supξ0∈Cjδ,3

∣

∣

∣D2H(ξ0)∣

∣

∣ b0 ξ2 =: b1 ξ2

such that when plugging in the definition of H (cf. lemma 2.3.5) the first component of (2.35) is boundedfrom below, i.e.

(F (ξ, ε) +G(t, ξ, ε))1 = ε ξ2 + ε2 ∆2(t, ξ, ε) ≥ ε ξ2 (1− ε b1) ≥ ε 12 ξ2

provided that ε is sufficiently small and 0 < ε < ε3. Hence for ξ ∈ T jε,3 ⊂ Cjδ,3 we have

(F (ξ, ε) +G(t, ξ, ε))1 ≥ ε 12 dist

(

Uj,+−,ε, ξjH

)

> 0.

By consequence every solution of (2.35) starting in T jε,3 reaches the border T j

ε,3 ∩ α × R within finitetime. A very similar argument leads to

(F (ξ, ε) +G(t, ξ, ε))1 < −ε 12 β

in T jε,5 such that solutions starting in T j

ε,5 leave this set at ξ2 = −αδ,m within finite time (after possibly

having passed the region T jε,2). As

sup

(F (ξ, ε) +G(t, ξ, ε))2

∣

∣

∣ξ ∈ T jε,4

< 0

we conclude that every solution in T jε,4 reaches T j

ε,5 within finite time (after possibly having passed through

T jε,1).

Summarizing these results on T jε,k, k = 1, . . . , 5 we conclude that every solution ξ(t) of (2.35) starting in

Ajδ ∩Dj

ε,δ eventually crosses the set ξ2 = −αδ,m indeed.

As mentioned before, the aim of this section is to show that in the case where |a0| <√

(ac1)2 + (as1)

2

the set of solutions of (2.25) not passing the inner region is at most of size O(ε). Using the notationintroduced before we now are in the position to formulate this statement in a precise way. This is thesubject of the following main result.


Proposition 2.3.11 There exists ε4 > 0 such that for every 0 < ε < ε4 the following statement holds:

The (Lebesgue) measure of the set Ajδ \(

Ajδ ∩ Dj

ε,δ

)

is of size O(ε). Thus the set of initial values for

which the corresponding solutions of (2.35) are possibly captured in the inner region |ξ2| ≤ αδ,m (andtherefore near the resonance h = hm of (2.25)) tends towards zero as ε→ 0.

PROOF: In order to prove the statement given, one shows that the point P1(ε) of intersection of Uj,+−,εwith Aj

δ and Q1(ε) of Uj+1,++,ε with Aj

δ are O(ε)–close to the points P1(0), Q1(0) i.e. the border of Ajδ

(cf. figure 2.7). By definition of the sets Ajδ, D

jε,δ this establishes the claim made in proposition 2.3.11.

We will give the proof for the more delicate situation of P1(ε). The analogous result on Q1(ε) then maybe proved using the same arguments.

Let us begin by recalling definition 2.40 of the vector field w. Plugging this into (2.43 b) yields

ξ = ε J∇H(ξ)− ε2 (1 + Λ)∆√

1− ε2 ∆2∇H(ξ).

Since (2.43 b) is autonomous, we may rescale the time variable t without any consequences on the stablemanifold Uj,+−,ε of ξjH . Thus let us consider the system

d

dτξ = J∇H(ξ)− α(ε)∇H(ξ) (2.50)

where α(ε) := ε (1 + Λ) ∆√1−ε2 ∆2

and τ = ε t.

Let T ∈ R2×2 denote the matrix such that T−1 JD2H(ξjH)T is diagonal and set ξ − ξjH = T x. Then(2.50) is transformed into

d

dτx = T−1

(

J∇H(ξjH + T x)− α(ε)∇H(ξjH + T x))

. (2.51)

We proceed in several steps :

1. For ε sufficiently small, system (2.51) admits (un)stable manifolds U+ε , U−

ε of the hyperbolic fixedpoint x = 0. It is evident that the stable manifold U+

ε of x = 0 with respect to (2.51) correspondsto the stable manifold Uj,+−,ε of ξjH with respect to (2.43 b).

2. Following the general results on the existence of invariant manifolds of hyperbolic fixed points it ispossible to locally represent the (un)stable manifolds U±

ε as graphs over a subset of the x1, x2–axisrespectively. More precisely there exist δ1 > 0, ε4 and functions s+, s− in C1([−δ1, δ1]× [0, ε4],R)such that for any 0 ≤ ε ≤ ε4 the following holds:

(x1, x2) | x2 = s+(x1, ε), |x1| ≤ δ1

⊂ U+ε

(x1, x2) | x1 = s−(x2, ε), |x2| ≤ δ1

⊂ U−ε .

3. Given a number 0 < δ2 ≤ 12 δ1 which will be fixed in step 5 we consider the intersection

Qδ2,ε :=

(x1, x2)∣

∣

∣

∣x1 − s−(x2, ε)∣

∣ ≤ δ2,∣

∣x2 − s+(x1, ε)∣

∣ ≤ δ2

of the δ2–neighbourhoods of the graphs of s+ and s−. As depicted in figure 2.10 we then definethe points P2(ε), P3(ε) of intersection of ∂Qδ2,ε with U+

ε as well as the (upper) point P4(ε) of

intersection of U−ε with ∂Qδ2,ε.


4. For ε = 0 there exists a homoclinic orbit which we will denote by xh. Let τ0 denote the real numbersuch that the second coordinate xh(τ0)2 of xh(τ0) is equal to δ2. Then the distance of the manifoldsU+ε and U−

ε near xh(τ0) is given by the expression

− (1 + Λ) ε ∆

∫

int xh

trace[

T−1D2H(ξjH + T x)T]

∣

∣

∣T−1 J∇H(ξjH + T xh(τ0))∣

∣

∣

dx+O(ε2)

(cf. formulae (4.5.6), (4.5.11) and (4.5.15) in [5]). We therefore conclude∣

∣

∣P3(ε)− P4(ε)∣

∣

∣ = O(ε).

Figure 2.10: The situation considered in the proof of proposition 2.3.11 (x–coordinates)


5. For 0 ≤ ε ≤ ε4 (where ε4 is to be chosen sufficiently small) there exists a further change ofcoordinates defined via

(

x1x2

)

=

(

x1x2

)

+

(

s−(x2, ε)s+(x1, ε)

)

. (2.52)

Let Pj(ε) denote the points corresponding to Pj(ε) in the new coordinates (j = 2, 3, 4). Then

P2(ε)1 = −δ2 and P3(ε)2 = δ2. Since∣

∣

∣P3(ε)− P4(ε)

∣

∣

∣= O(ε) we derive

∣

∣P3(ε)− P4(ε)∣

∣ = O(ε) and

taking into account that P4(ε)1 = 0 therefore find P3(ε)1 to be negative and of size O(ε).

Figure 2.11: The situation considered in the proof of proposition 2.3.11 (x–coordinates)

Applying the change (2.52) from x to x coordinates on (2.51) then yields a system of the form

d

dτx1 = −x1 (λ1(ε) + x1 g1(x, ε) + x2 g2(x, ε))

d

dτx2 = x2 (λ2(ε) + x1 g3(x, ε) + x2 g4(x, ε)) ,

(2.53)

where 0 < λ1(0) = λ2(0) =: λ0 and ε4 may be chosen sufficiently small such that λ1(ε), λ2(ε) arepositive for all 0 ≤ ε ≤ ε4. Taking the supremum over |x1| ≤ 1

2δ1, |x2| ≤ 12δ1 and 0 ≤ ε ≤ ε4 we

find a bound b1 > 0 of the maps gj (j = 1, . . . , 4).

The aim of this step is to establish that the point P2(ε) satisfies P2(ε)2 = O(ε). The strategyherefore consist in explicitely finding an appropriate negative invariant set for (2.53) containingP3(ε). This is achieved by considering the orbits of the equation

d

dτx =

(

−x1 (λ1(ε)− β x1)x2 (λ2(ε)− β x2)

)

(2.54)


where the constant β is set equal to 10 b1. We then find

(

−x1 (λ1(ε)− β x1)x2 (λ2(ε)− β x2)

)

∧(

−x1 (λ1(ε) + x1 g1(x, ε) + x2 g2(x, ε))x2 (λ2(ε) + x1 g3(x, ε) + x2 g4(x, ε))

)

= −x1 x2[

(λ1(ε)− β x1) (λ2(ε) + x1 g3(x, ε) + x2 g4(x, ε))

− (λ2(ε)− β x2) (λ1(ε) + x1 g1(x, ε) + x2 g2(x, ε))]

= −x1 x2[

− x1 (β λ0 + λ0 g1(x, ε)− λ0 g3(x, ε) + β x1 g3(x, ε) + β x2 g4(x, ε) +O(ε))

+ x2 (β λ0 + λ0 g4(x, ε)− λ0 g2(x, ε) + β x2 g2(x, ε) + β x1 g1(x, ε) +O(ε))]

and hence for δ2 := min

12 δ1,

λ0

10 b1

and −δ2 ≤ x1 ≤ 0, 0 ≤ x2 ≤ δ2

≥ −x1 x2[

− x1 (β λ0 − 2λ0 b1 − 2 β δ2 b1 +O(ε)) + x2 (β λ0 − 2λ0 b1 − 2 β δ2 b1 +O(ε))]

.

We therefore obtain(

−x1 (λ1(ε)− β x1)x2 (λ2(ε)− β x2)

)

∧(

−x1 (λ1(ε) + x1 g1(x, ε) + x2 g2(x, ε))x2 (λ2(ε) + x1 g3(x, ε) + x2 g4(x, ε))

)

≥ −x1 x2β λ02

(x2 − x1) (2.55)

provided that ε4 is chosen suitably small and 0 ≤ ε ≤ ε4.

Solving (2.54) explicitely then implies that the point P2(ε) satisfying P2(ε)1 = −δ2 and lying on

the orbit γ (cf. figure 2.11) through P3(ε) satisfies P2(ε)2 =(

−P3(ε)1)λ2(0)/λ1(0)

(const+O(ε)) anddue to λ2(0) = λ1(0) therefore is O(ε) close to the x1–axis, indeed. By consequence of (2.55) thepoint P2(ε) must be located between the orbit γ and the x1–axis. Thus P2(ε)2 = O(ε).

Changing back to x–coordinates we find dist(

P2(ε), graph (s+( . , ε))

)

= O(ε). As s+ ∈ C1 there

exists a constant b2 > 0 such that |s+(x1, ε)− s+(x1, 0)| ≤ b2 ε uniformly with respect to |x1| ≤ δ1.

This argument eventually is used to establish that dist(

P2(ε), graph (s+( . , 0))

)

= O(ε) which

together with graph (s+( . , 0)) ⊂ U+0 implies dist

(

P2(ε), U+0

)

= O(ε).

6. It remains to show that∣

∣

∣P1(ε)− P1(0)∣

∣

∣ = O(ε), i.e. P1(ε) is O(ε) far of the border of Ajδ (in x–

coordinates). However, as the vector field is bounded from below on the corresponding domain, thismay be established by comparing the distance of the trajectories of the solutions x(.; 0, P2(ε), ε)and x(.; 0, P2(ε), 0) of (2.51) passing through P2(ε).

7. Reversing the transformation ξ−ξjH = T x we see that the points P1(ε), P1(0) depicted in figure 2.7

(in ξ–coordinates) are identical to P1(ε), P1(0) (expressed in x–coordinates). As the transformationξ ↔ x applied is affine we eventually obtain |P1(ε)− P1(0)| = O(ε) as it had to be shown.


2.3.5 On the Proof of Existence of Capture in Resonance

In contrast to the preceeding subsections we now aim on the existence of solutions which do not pass theresonance zones. We will see that the criteria necessary to decide whether such solutions exist or not arebased on the discussion of the O(ε3)–terms in (2.1). The explicit determination of these terms for theapplication considered in chapter 4 requires an extensive amount of preparations and evaluations beyondthe scope of this work, however. Hence we content ourselves with a short sketch of the process necessaryto obtain these terms.

1. Consider the representation (2.29) of system (2.25) in the inner variables. We then are in thesituation dealt with in [14]. In this paper the author shows that the leading term of the Melnikovfunction is given by

d1 :=

∫

R

(

hs(s)a0 +

∑

l∈N∗

acl cos(lψs(s)) + asl sin(lψs(s))

)

∧(

f2,0(ψs(s), hs(s))

g2,0(ψs(s), hs(s))

)

ds

where f2,0(ψ, h), g

2,0(ψ, h) denote the O(1)–terms of the mean values of f2(t, ψ, h, ε), g2(t, ψ, h, ε)

with respect to t and (ψs, hs) is the homoclinic solution of the ”unperturbed system”3. (If system(2.29) is averaged up to O(ε3)–terms by using a near–identity transformation of the form (ψ, h) =(ψ, h)+ ε2w(t, ψ, h) first, this result may be understood as the well known Melnikov formula of theresulting system in (ψ, h)–coordinates.)

2. From the results given in [14] it may be seen at once that if d1 6= 0, the sign of d1 determinesthe orientation of the (time–dependant) stable and unstable invariant manifolds of the hyperbolic,km 2π–periodic solutions of (2.29).

3. d1 > 0:

If d1 is positive then the every section of the unsta-ble manifold with t = const lies ”outside” the stablemanifold and no capture into resonance is possible.

4. d1 < 0:

If d1 is negative, solutions may be caught in the areabetween the (time–dependant) stable and unstablemanifolds, ending inside the ”eye–shaped” region.Hence a capture into resonance is possible.

5. From definition 2.3.2 and lemma 2.3.3 we see that in order to find explicit formulae for the quantitiesf2,0(ψ, h) and g

2,0(ψ, h), a sufficiently explicit representation of the transformation vI(t, ϕ, h, ε) and

the coefficient maps f2k,n g3k,n is needed. From the representation (1.156) of the reduced system

it follows that the deduction of the quantities f2,0(ψ, h) and g2,0(ψ, h) requires in particular F2,1

k,n,3,

F3,0k,n,3 and S2

k,n(h). From definition 1.6.5, proposition 1.6.7 and (1.143), (1.138) we thus see thatthe explicit formulae (1.20), (1.21), (1.91), (1.92) must be evaluated when applying the theory tothe example of a miniature synchronous motor in chapter 4.

For the application considered in chapter 4 the procedure corresponding to this last step 5 requires asignificantly more extensive amount of preparations than in section 4.3. Although the questions in relationto the capture into resonance are of interest, we will omit this discussion in our application.

3i.e. system (2.29) omitting O(ε2)–terms.

Chapter 3

The Stability of the Set h = 0 inAction Angle Coordinates



The systems considered in this chapter are of the general form


h = P(h)ddhP(h)

g,1(t, ϕ, ε) + P(h)2

ddhP(h)

g,2(t, ϕ, ε) + P(h)3

ddhP(h)

g,3(t, ϕ,P(h), ε).(3.1)

Hereafter the assumptions listed in the following section are assumed to be true.



GA 3.1. Ω0 6∈ 12Z.

GA 3.2. The mappings f ,l, g,l are of class BCr (r ≥ 3) with respect to all arguments t, ϕ, r ∈ R, |ε| < ε1,2π–periodic with respect to t, ϕ and vanish for ε = 0.

GA 3.3. There exist maps f ,00 , f ,0s , f ,0c and g,10 in BC r(R× (−ε1, ε1),R), 2π–periodic with respect to t suchthat the following representation of f ,0, g,1 holds :

f ,0(t, ϕ, ε) = f ,00 (t, ε) + f ,0c (t, ε) cos(2ϕ) + f ,0s (t, ε) sin(2ϕ)

g,1(t, ϕ, ε) = g,10 (t, ε)− f ,0s (t, ε) cos(2ϕ) + f ,0c (t, ε) sin(2ϕ).(3.2)

GA 3.4. The maps f ,1, g,2 are π–anti–periodic1.

1cf. definition 1.6.13

111

112 Chapter 3. The Stability of the Set h = 0 in Action Angle Coordinates

GA 3.5. The function P is Cω on R, ddhP(h) > 0 for h 6= 0 and P(0) = 0. The derivatives dk

dhkP(h),1 ≤ k ≤ 4 are bounded uniformly.

Remark 3.1.1 If the General Assumptions GA1 of chapter 1 are fulfilled, then the reduced system (1.160)satisfies GA3. In particular lemma 1.6.12 and lemma 1.6.15 imply the properties assumed in GA 3.3 andGA 3.4.

3.2 Averaging the Linear Term

The aim of this section is to introduce a near–identical transformation for the action angle coordi-nates (ϕ, h) such that the linearization of the transformed equation of the action–variable is autonomous.The first proposition provides the result needed in a general form:

Proposition 3.2.1 Consider the following truncated system of (3.1):

ϕ = Ω0 + f ,0(t, ϕ, ε)

h = P(h)ddhP(h)

g,1(t, ϕ, ε).(3.3)

Then there exists ε2 > 0 and functions u, v (where u(t, ψ, 0) = 0 and v(t, 0) = 1), 2π–periodic withrespect to t and ψ, such that for every |ε| < ε2 the following statements hold:

1. The change of coordinates given by

ϕ = ψ + u(t, ψ, ε)

P(h) = rv(t, ε)

√

1 + ∂ψu(t, ψ, ε)

(3.4)

is well defined and transforms the system (3.3) into

ψ = Ω(ε)

r = r g,10,0(ε)(3.5)

where the continuous function Ω fulfills Ω(0) = Ω0 and g,10,0(ε) is the mean value of the map g,1 in(3.3) i.e.

g,10,0(ε) =1

(2π)2

2π∫

0

2π∫

0

g,1(t, ϕ, ε) dt dϕ. (3.6)

2. The map u solves the partial differential equation

∂tu(t, ψ, ε) + (1 + ∂ψu(t, ψ, ε)) Ω(ε) = Ω0 + f ,0(t, ψ + u(t, ψ, ε), ε, a) (3.7)

and v satisfies the linear equation:

d

dtv(t, ε) =

(

g,10 (t, ε)− g,10,0(ε))

v(t, ε). (3.8)

3.2. Averaging the Linear Term 113

PROOF: The idea of this proof is to derive a system in cartesian coordinates equivalent to (3.3) andthen applying standard results of Floquet theory. As it will be seen such a system may be found due tothe special form of the truncated vector field as assumed in GA 3.3. We proceed in the following steps:

1. Define the time–dependent matrix

M(t, ε) :=

g,10 (t, ε) + f ,0s (t, ε) 1Ω0

(

f ,0c (t, ε) + f ,00 (t, ε))

Ω0

(

f ,0c (t, ε)− f ,00 (t, ε))

g,10 (t, ε)− f ,0s (t, ε)

as well as R :=

[

0 1−Ω2

0 0

]

and introduce the cartesian coordinates x := P(h)

(

1Ω0

sin(ϕ)

cos(ϕ)

)

. From

(3.3) it then is found that

x =

[P(h) 1Ω0

cos(ϕ) ddhP(h) 1

Ω0sin(ϕ)

−P(h) sin(ϕ) ddhP(h) cos(ϕ)

] (

ϕ

h

)

=

[P(h) 1Ω0

cos(ϕ) ddhP(h) 1

Ω0sin(ϕ)

−P(h) sin(ϕ) ddhP(h) cos(ϕ)

]

(

Ω0 + f ,0(t, ϕ, ε)P(h)ddhP(h)

g,1(t, ϕ, ε)

)

= P(h)

[

0 1−Ω2

0 0

] (

1Ω0

sin(ϕ)

cos(ϕ)

)

+P(h)

(

f ,0(t, ϕ, ε)

(

1Ω0

cos(ϕ)

− sin(ϕ)

)

+ g,1(t, ϕ, ε)

(

1Ω0

sin(ϕ)

cos(ϕ)

))

.

Plugging in the representation (3.2) assumed in GA 3.3 yields

f ,0(t, ϕ, ε)

(

1Ω0

cos(ϕ)

− sin(ϕ)

)

+ g,1(t, ϕ, ε)

(

1Ω0

sin(ϕ)

cos(ϕ)

)

= f ,00 (t, ε)

(

1Ω0

cos(ϕ)

− sin(ϕ)

)

+ f ,0s (t, ε)

(

sin(2ϕ) 1Ω0

cos(ϕ)− cos(2ϕ) 1Ω0

sin(ϕ)

− sin(2ϕ) sin(ϕ)− cos(2ϕ) cos(ϕ)

)

+g,10 (t, ε)

(

1Ω0

sin(ϕ)

cos(ϕ)

)

+ f ,0c (t, ε)

(

cos(2ϕ) 1Ω0

cos(ϕ) + sin(2ϕ) 1Ω0

sin(ϕ)

− cos(2ϕ) sin(ϕ) + sin(2ϕ) cos(ϕ)

)

= f ,00 (t, ε)

(

1Ω0

cos(ϕ)

− sin(ϕ)

)

+ f ,0s (t, ε)

(

1Ω0

sin(ϕ)

− cos(ϕ)

)

+g,10 (t, ε)

(

1Ω0

sin(ϕ)

cos(ϕ)

)

+ f ,0c (t, ε)

(

1Ω0

cos(ϕ)

sin(ϕ)

)

= M(t, ε)

(

1Ω0

sin(ϕ)

cos(ϕ)

)

hence (3.3) is equivalent to

x = (R+M(t, ε)) x. (3.9)

2. For ε = 0 the monodromy operator of (3.9) is given by exp (2πR). The Floquet multipliers µ1(0),µ2(0) of exp (2πR) are given by µ1,2(0) = e±i 2πΩ0 . Since Ω0 6∈ 1

2Z (cf. GA 3.1) the multipliersµ1(0), µ2(0) are therefore non–real, complex conjugate numbers. As the dependence of the mon-odromy matrix on the parameter ε is continuous it follows that for ε 6= 0 the corresponding Floquetmultipliers µ1(ε), µ2(ε) of (3.9) are given by two different complex conjugate numbers

µ1,2(ε) = λ(ε) e±i 2π Ω(ε)


(provided that |ε| < ε2, ε2 > 0 sufficiently small) where λ, Ω depend continuously on ε and λ(0) = 1,Ω(0) = Ω0.

Hence we are in the position to apply standard results of Floquet theory, as for instance given inthe proof of Lemma 4, p. 270 in [11]. By consequence of this theory there exists a 2π–periodic

transformation T : R× (−ε2, ε2) → R2×2 of class BC r with T (t, 0) =

[

1Ω0

0

0 1

]

such that setting

x = T (t, ε) y (3.10)

the equation (3.9) is transformed to

y = B(ε) y (3.11)

where B(ε) :=

[

12π ln(λ(ε)) Ω(ε)

−Ω(ε) 12π ln(λ(ε))

]

.

3. The next step consist in transforming (3.11) back into action angle coordinates: define the coordi-

nates (ψ, r) via y = r

(

sin(ψ)cos(ψ)

)

and calculate

(

ψr

)

=

[

r cos(ψ) sin(ψ)−r sin(ψ) cos(ψ)

]−1

×[

12π ln(λ(ε)) Ω(ε)

−Ω(ε) 12π ln(λ(ε))

] (

r sin(ψ)r cos(ψ)

)

=1

2πln(λ(ε))

(

0r

)

+ Ω(ε)

(

10

)

.

Taking into account that by consequence of Liouville’s theorem (e.g. (11.4) in [1])

λ(ε)2 = µ1(ε)µ2(ε) = exp

2π∫

0

trace (R +M(t, ε)) dt

we find

1

2πln(λ(ε)) =

1

2π

2π∫

0

g,10 (t, ε) dt = g,10,0(ε) (3.12)

such that

ψ = Ω(ε)

r = r g,10,0(ε)

This corresponds to the representation claimed in (3.5).

4. In order to establish the first statement completely, it remains to show that the change of coordinatescarried out in the first three steps may be expressed as in (3.4).

Summarizing these transformations we have

P(h)

(

1Ω0

sin(ϕ)

cos(ϕ)

)

= r T (t, ε)

(

sin(ψ)cos(ψ)

)

,


u(t, , )εψ

εU(t, )

ψϕη

P(h)

r

ξ

Figure 3.1: Illustration of (3.14)

and (left–) multiplication with T−1(t, 0) =

[

Ω0 00 1

]

yields

P(h)

(

sin(ϕ)cos(ϕ)

)

= r U(t, ε)

(

sin(ψ)cos(ψ)

)

(3.13)

where U(t, ε) := T−1(t, 0)T (t, ε) satisfies U(t, 0) = IR2 . As illustrated in figure 3.1 it may be seenthat setting

(

ξ(t, ψ, ε)η(t, ψ, ε)

)

:= U(t, ε)

(

sin(ψ)cos(ψ)

)

the identities

P(h) = r√

ξ(t, ψ, ε)2 + η(t, ψ, ε)2

ϕ = arg(η(t, ψ, ε) + i ξ(t, ψ, ε))(3.14)

hold. Taking into account that U(t, 0) = IR2 implies ξ(t, ψ, 0) = sin(ψ) and η(t, ψ, 0) = cos(ψ) weconclude that there exists a map u ∈ BC r(R2 × (−ε2, ε2),R) (which is 2π–periodic with respectto t and ψ) satisfying u(t, ψ, 0) = 0 such that

ϕ = ψ + u(t, ψ, ε). (3.15)

This corresponds to the representation given for ϕ in (3.4). Taking derivatives with respect to ψ ittherefore follows from (3.14) and (3.15) that

1 + ∂ψu(t, ψ, ε) = ddψ arg(η(t, ψ, ε) + i ξ(t, ψ, ε))

=1

1 +(

ξ(t,ψ,ε)η(t,ψ,ε)

)2

∂ψξ(t, ψ, ε) η(t, ψ, ε)− ξ(t, ψ, ε) ∂ψη(t, ψ, ε)

η(t, ψ, ε)2

=∂ψξ(t, ψ, ε) η(t, ψ, ε)− ξ(t, ψ, ε) ∂ψη(t, ψ, ε)

ξ(t, ψ, ε)2 + η(t, ψ, ε)2,

hence

√

ξ(t, ψ, ε)2 + η(t, ψ, ε)2 =

√

∂ψξ(t, ψ, ε) η(t, ψ, ε)− ξ(t, ψ, ε) ∂ψη(t, ψ, ε)√

1 + ∂ψu(t, ψ, ε). (3.16)


Using

(

∂ψξ(t, ψ, ε)∂ψη(t, ψ, ε)

)

= U(t, ε)

(

cos(ψ)− sin(ψ)

)

= U(t, ε)J

(

sin(ψ)cos(ψ)

)

we find

v(t, ε) :=√

∂ψξ(t, ψ, ε) η(t, ψ, ε)− ξ(t, ψ, ε) ∂ψη(t, ψ, ε)

=

√

(

U(t, ε)J

(

sin(ψ)cos(ψ)

)∣

∣

∣

∣

J U(t, ε)

(

sin(ψ)cos(ψ)

))

=√

detU(t, ε). (3.17)

Combining (3.14) and (3.16) yields

P(h) = rv(t, ε)

√

1 + ∂ψu(t, ψ, ε)(3.18)

as claimed in (3.4).

5. It remains to establish the identities (3.7) and (3.8) claimed in the second assertion. Taking deriva-tives of (3.15) with respect to t it follows for every solution (ϕ, r) of (3.3), (ψ, r) of (3.5) respectivelythat

Ω0 + f ,0(t, ψ + u(t, ψ, ε), ε, a) = ϕ = ∂tu(t, ψ, ε) + (1 + ∂ψu(t, ψ, ε)) Ω(ε)

hence (3.7). In order to establish (3.8) we derive

x = (R+M(t, ε)) T (t, ε) y

from (3.9), (3.10) while on the other hand taking derivatives in (3.10) implies

x = (∂tT (t, ε) + T (t, ε)B(ε)) y

leading to

∂tT (t, ε) =[

R+M(t, ε)− T (t, ε)B(ε)T−1(t, ε)]

T (t, ε). (3.19)

By consequence of Liouville’s theorem and (3.12) we have

d

dtdet T (t, ε) = trace

[

R+M(t, ε)− T (t, ε)B(ε)T−1(t, ε)]

detT (t, ε)

= (trace (R+M(t, ε))− traceB(ε)) detT (t, ε)

=

(

2 g,10 (t, ε)− 21

2πln(λ(ε))

)

detT (t, ε)

= 2

g,10 (t, ε)− 1

2π

2π∫

0

g,10 (t, ε) dt

detT (t, ε).

Since detT−1(t, 0) = Ω0 we conclude from (3.8), (3.17) and the definition of U(t, ε) that

d

dtv(t, ε) =

ddt detU(t, ε)

2√

detU(t, ε)=

ddt (Ω0 detT (t, ε))

2√

Ω0 detT (t, ε)

=√

Ω0

g,10 (t, ε)− 1

2π

2π∫

0

g,10 (t, ε) dt

√

detT (t, ε)

=(

g,10 (t, ε)− g,10,0(ε))

v(t, ε)

and therefore have established (3.8) as well.


This accomplishes the proof of proposition 3.2.1.

In a next step we apply the transformation given by proposition 3.2.1 on the full system (3.1) instead ofthe truncated system (3.3) leading to the main result of this section.

Corollary 3.2.2 Applying the change of coordinates (3.4) given by proposition 3.2.1 to


h = P(h)ddhP(h)

g,1(t, ϕ, ε) + P(h)2

ddhP(h)

g,2(t, ϕ, ε) + P(h)3

ddhP(h)

g,3(t, ϕ,P(h), ε),(3.1)

yields the system

ψ = Ω(ε) + r f ,1(t, ψ, ε) + r2 f ,2(t, ψ, r, ε)

r = r g,10,0(ε) + r2 g,2(t, ψ, ε) + r3 g,3(t, ψ, r, ε)(3.20)

where

f ,1(t, ψ, ε) = f ,1(t, ψ + u(t, ψ, ε), ε)v(t, ε)

√

1 + ∂ψu(t, ψ, ε)3

f ,2(t, ψ, r, ε) = f ,2(t, ψ + u(t, ψ, ε), r v(t,ε)√1+∂ψu(t,ψ,ε)

, ε)v(t, ε)2

(1 + ∂ψu(t, ψ, ε))2

g,2(t, ψ, ε) = f ,1(t, ψ, ε)∂2ψu(t, ψ, ε)

2 (1 + ∂ψu(t, ψ, ε))+ g,2(t, ψ + u(t, ψ, ε), ε)

v(t, ε)√

1 + ∂ψu(t, ψ, ε)

g,3(t, ψ, r, ε) = f ,2(t, ψ, r, ε)∂2ψu(t, ψ, ε)

2 (1 + ∂ψu(t, ψ, ε))+ g,3(t, ψ + u(t, ψ, ε), r v(t,ε)√

1+∂ψu(t,ψ,ε), ε)

v(t, ε)2

1 + ∂ψu(t, ψ, ε)(3.21)

are of class BC r−1.

PROOF: Taking the derivative of the first equation in (3.4) yields

ϕ = ∂tu(t, ψ, ε) + (1 + ∂ψu(t, ψ, ε)) ψ,

while on the other hand (3.1) implies

ϕ = Ω0 + f ,0(t, ψ + u(t, ψ, ε), ε) + r v(t,ε)√1+∂ψu(t,ψ,ε)

f ,1(t, ψ + u(t, ψ, ε), ε)

+ r2 v(t,ε)2

1+∂ψu(t,ψ,ε)f ,2(t, ψ + u(t, ψ, ε), r v(t,ε)√

1+∂ψu(t,ψ,ε), ε).

Therefore

ψ =Ω0 + f ,0(t, ψ + u(t, ψ, ε), ε)− ∂tu(t, ψ, ε)

1 + ∂ψu(t, ψ, ε)+ r v(t,ε)√

1+∂ψu(t,ψ,ε)3 f

,1(t, ψ + u(t, ψ, ε), ε)

+ r2 v(t,ε)2

(1+∂ψu(t,ψ,ε))2 f

,2(t, ψ + u(t, ψ, ε), r v(t,ε)√1+∂ψu(t,ψ,ε)

, ε)


and since (3.7) implies

Ω0 + f ,0(t, ψ + u(t, ψ, ε), ε)− ∂tu(t, ψ, ε)

1 + ∂ψu(t, ψ, ε)= Ω(ε)

we conclude

ψ = Ω(ε) + r v(t,ε)√1+∂ψu(t,ψ,ε)

3 f,1(t, ψ + u(t, ψ, ε), ε) + r2 v(t,ε)2

(1+∂ψu(t,ψ,ε))2 f

,2(t, ψ + u(t, ψ, ε), r v(t,ε)√1+∂ψu(t,ψ,ε)

, ε)

= Ω(ε) + r f ,1(t, ψ, ε) + r2 f ,2(t, ψ, r, ε).

Taking the derivative of the second equation in (3.4) yields

r = P(h)d

dt

(

√

1 + ∂ψu(t, ψ, ε)

v(t, ε)

)

+ ddhP(h)

√

1 + ∂ψu(t, ψ, ε)

v(t, ε)h

= P(h)∂t∂ψu(t, ψ, ε) + ∂2ψu(t, ψ, ε) ψ

2√

1 + ∂ψu(t, ψ, ε) v(t, ε)− P(h)

ddtv(t, ε)

v(t, ε)

√

1 + ∂ψu(t, ψ, ε)

v(t, ε)

+ ddhP(h)

√

1 + ∂ψu(t, ψ, ε)

v(t, ε)h

= P(h)

ddψ

(

∂tu(t, ψ, ε) + ∂ψu(t, ψ, ε) Ω(ε))

+ ∂2ψu(t, ψ, ε)(

r f ,1(t, ψ, ε) + r2 f ,2(t, ψ, r, ε))

2√

1 + ∂ψu(t, ψ, ε) v(t, ε)

−P(h)ddtv(t, ε)

v(t, ε)

√

1 + ∂ψu(t, ψ, ε)

v(t, ε)+ d

dhP(h)

√

1 + ∂ψu(t, ψ, ε)

v(t, ε)h.

From GA 3.3 we deduce the identity

12∂ϕf

,0(t, ϕ, ε) = g,10 (t, ε)− g,1(t, ϕ, ε)

which together with (3.7) implies

r = P(h)

ddψ

(

Ω0 − Ω(ε) + f ,0(t, ψ + u(t, ψ, ε), ε))

2√

1 + ∂ψu(t, ψ, ε) v(t, ε)− P(h)

ddtv(t, ε)

v(t, ε)

√

1 + ∂ψu(t, ψ, ε)

v(t, ε)

+P(h)∂2ψu(t, ψ, ε)

(

r f ,1(t, ψ, ε) + r2 f ,2(t, ψ, r, ε))

2√

1 + ∂ψu(t, ψ, ε) v(t, ε)+ d

dhP(h)

√

1 + ∂ψu(t, ψ, ε)

v(t, ε)h

= P(h)∂ϕf

,0(t, ψ + u(t, ψ, ε), ε) (1 + ∂ψu(t, ψ, ε))

2√

1 + ∂ψu(t, ψ, ε) v(t, ε)− P(h)

ddtv(t, ε)

v(t, ε)

√

1 + ∂ψu(t, ψ, ε)

v(t, ε)

+P(h)∂2ψu(t, ψ, ε)

(

r f ,1(t, ψ, ε) + r2 f ,2(t, ψ, r, ε))

2√

1 + ∂ψu(t, ψ, ε) v(t, ε)+ d

dhP(h)

√

1 + ∂ψu(t, ψ, ε)

v(t, ε)h

= P(h)

[

g,10 (t, ε)− g,1(t, ψ + u(t, ψ, ε), ε)]

(1 + ∂ψu(t, ψ, ε))√

1 + ∂ψu(t, ψ, ε) v(t, ε)− P(h)

ddtv(t, ε)

v(t, ε)

√

1 + ∂ψu(t, ψ, ε)

v(t, ε)

+P(h)∂2ψu(t, ψ, ε)

(

r f ,1(t, ψ, ε) + r2 f ,2(t, ψ, r, ε))

2√

1 + ∂ψu(t, ψ, ε) v(t, ε)+ d

dhP(h)

√

1 + ∂ψu(t, ψ, ε)

v(t, ε)h


and plugging in the explicit form for h as in (3.1) together with (3.8), (3.18) finally implies

r = P(h)[

g,10 (t, ε)− g,1(t, ψ + u(t, ψ, ε), ε)]

√

1 + ∂ψu(t, ψ, ε)

v(t, ε)− P(h)

ddtv(t, ε)

v(t, ε)

√

1 + ∂ψu(t, ψ, ε)

v(t, ε)

+P(h)∂2ψu(t, ψ, ε)

(

r f ,1(t, ψ, ε) + r2 f ,2(t, ψ, r, ε))

2√

1 + ∂ψu(t, ψ, ε) v(t, ε)

+P(h)

√

1 + ∂ψu(t, ψ, ε)

v(t, ε)g,1(t, ψ + u(t, ψ, ε), ε)

+

√

1 + ∂ψu(t, ψ, ε)

v(t, ε)

(

P(h)2g,2(t, ψ + u(t, ψ, ε), ε) + P(h)

3g,3(t, ψ + u(t, ψ, ε),P(h), ε)

)

= r

(

g,10 (t, ε)−ddtv(t, ε)

v(t, ε)

)

+ r2

(

∂2ψu(t, ψ, ε) f,1(t, ψ, ε)

2 (1 + ∂ψu(t, ψ, ε))+g,2(t, ψ + u(t, ψ, ε), ε) v(t, ε)

√

1 + ∂ψu(t, ψ, ε)

)

+r3

∂2ψu(t, ψ, ε) f,2(t, ψ, r, ε)

2 (1 + ∂ψu(t, ψ, ε))+g,3(t, ψ + u(t, ψ, ε), r v(t,ε)√

1+∂ψu(t,ψ,ε), ε) v(t, ε)2

1 + ∂ψu(t, ψ, ε)

= r(

g,10 (t, ε)−(

g,10 (t, ε)− g,10,0(ε)))

+ r2 g,2(t, ψ, ε) + r3 g,3(t, ψ, r, ε).

In view of the definition (3.6) we see that this establishes the claim given in corollary 3.2.2.

Note that since u ∈ BC r the right hand side of (3.4) is of class BC r−1 provided that |ε| is sufficientlysmall. Without any loss of generality we may assume that this is the case if |ε| < ε2. Hence it may beshown that the maps defined in (3.21) are BC r−1.

The next corollary deals with the case of linear stability of the invariant set h = 0 of (3.1).

Corollary 3.2.3 The form (3.20) achieved in this section admits the following conclusion :

1. Assume that there exists a positive constant r∞ such that for all 0 ≤ r ≤ r∞ and any |ε| < ε2 theinequality

∣

∣

∣g,10,0(ε)

∣

∣

∣ > r∣

∣g,2(t, ψ, ε) + r g,3(t, ψ, r, ε)∣

∣ ∀ t, ϕ ∈ R (3.22)

is fulfilled. Then if g,10,0(ε) < 0 the invariant set r = 0 of (3.20) and hence the invariant seth = 0 of (3.1) is stable and the set (−r∞, r∞) is contained in the domain of attraction ofr = 0. If on the other hand g,10,0(ε) > 0 is true then r = 0 is unstable and every non–trivialsolution in [−r∞, r∞] leaves [−r∞, r∞].

2. Consider the situation where f ,j, g,j are of order O(ε2). From (3.7) and (3.8) one then mayconclude that u(t, ψ, ε) = O(ε2), v = 1+O(ε2). Together with the identities given in (3.21) we thenfind g,2(t, ψ, ε) = O(ε2), g,3(t, ψ, r, ε) = O(ε2). Hence if g,10,0(ε) 6= 0 then there exists a constant r∞such that the estimate (3.22) is satisfied. Note that in order to discuss the stability of h = 0 itthen suffices to consider the sign of the O(ε2)–terms g2,10,0 of g,10,0(ε). In particular the mappings uand v are not needed explicitely.

This result may be established in a similar way as shown in the proof of proposition 2.3.1.


3.3 The Mean Value of the Quadratic Term

In this section we show that by consequence of GA 3.3 and GA 3.4 the mean value of g,2 with respect toψ is zero.

Lemma 3.3.1 The map u introduced in proposition 3.2.1 is π–periodic with respect to ψ.

PROOF: The proof of this lemma proceeds in a very similar way to the proof of lemma 1.6.15. Takingthe identities (3.2), (3.7) and splitting u into the π–periodic and π–anti–periodic part with respect toψ, i.e.

u(t, ψ, ε) = u+(t, ψ, ε) + u−(t, ψ, ε)

yields the equations

∂tu+(t, ψ, ε) + ∂tu−(t, ψ, ε) +(

1 + ∂ψu+(t, ψ, ε) + ∂ψu−(t, ψ, ε))

Ω(ε)

= Ω0 + f ,00 (t, ε)

+f ,0c (t, ε) cos(2 (ψ + u+(t, ψ, ε) + u−(t, ψ, ε))) + f ,0s (t, ε) sin(2 (ψ + u+(t, ψ, ε) + u−(t, ψ, ε)))

= Ω0 + f ,00 (t, ε)

+f ,0c (t, ε)[

cos(2ψ) cos(2 u+(t, ψ, ε)) cos(2 u−(t, ψ, ε))− cos(2ψ) sin(2 u+(t, ψ, ε)) sin(2 u−(t, ψ, ε))

− sin(2ψ) sin(2 u+(t, ψ, ε)) cos(2 u−(t, ψ, ε))− sin(2ψ) cos(2 u+(t, ψ, ε)) sin(2 u−(t, ψ, ε))]

+f ,0s (t, ε)[

sin(2ψ) cos(2 u+(t, ψ, ε)) cos(2 u−(t, ψ, ε))− sin(2ψ) sin(2 u+(t, ψ, ε)) sin(2 u−(t, ψ, ε))

+ cos(2ψ) sin(2 u+(t, ψ, ε)) cos(2 u−(t, ψ, ε)) + cos(2ψ) cos(2 u+(t, ψ, ε)) sin(2 u−(t, ψ, ε))]

.

Writing down the π–anti–periodic part one then finds

∂tu−(t, ψ, ε) + ∂ψu−(t, ψ, ε) Ω(ε)

= f ,0c (t, ε)[

− cos(2ψ) sin(2 u+(t, ψ, ε))− sin(2ψ) cos(2 u+(t, ψ, ε))]

sin(2 u−(t, ψ, ε))

+f ,0s (t, ε)[

− sin(2ψ) sin(2 u+(t, ψ, ε)) + cos(2ψ) cos(2 u+(t, ψ, ε))]

sin(2 u−(t, ψ, ε))

Since u−(t, ψ, ε) := 0 is the unique solution of this last equation (cf. section 4.7.2), we have

u(t, ψ, ε) = u+(t, ψ, ε),

i.e. u(t, ψ, ε) is π–periodic.

3.3. The Mean Value of the Quadratic Term 121

We now are in the position to prove the main result of this subsection.

Proposition 3.3.2 The map g,2 (as defined in corollary 3.2.2) is π–anti–periodic with respect to ψ andtherefore has the mean value zero.

PROOF: It is easy to see that since u(t, ψ, ε) is π–periodic, the same holds for ∂ψu(t, ψ, ε) and∂2ψu(t, ψ, ε). This implies as well that if ε is sufficiently small then the map ψ 7→ (1 + ∂ψu(t, ψ, ε))

α

is defined and π–periodic for α ∈ −1,− 32 ,− 1

2.

As u(t, ψ, ε) is π–periodic and f ,1, g,2 are π–anti–periodic (GA 3.4), the maps ψ 7→ f ,1(t, ψ+u(t, ψ, ε), ε),ψ 7→ g,2(t, ψ + u(t, ψ, ε), ε) are π–anti–periodic functions as well.

From (3.21) we find f ,1 to be π–anti–periodic with respect to ψ. This finally implies that g,2 must beπ–anti–periodic as well.

As the mean value of a π–anti–periodic map is zero (cf. remark 1.6.14) the proof of proposition 3.3.2 iscomplete.


3.4 Averaging the Quadratic and Cubic Term

In this last section of chapter 3 we consider the situation where g,10,0(ε) = 0 in (3.20). Thus the invariantset r = 0 may not be linearly stable or unstable. Aiming on the discussion of a possible algebraic(in)stability we will apply a further near–identity transformation on the action variable r. By consequenceof the results found in section 3.3 we will find a representation of (3.20) where the leading r–term of ris autonomous and of order r3 instead of the non–autonomous representation of order r2 in (3.20). Thetransformation applied is constructed by the standard way of averaging techniques. As we have usedthese methods in the previous chapter the proofs given here are not carried out in detail.

Lemma 3.4.1 There exist positive constants r∞ and ε3 as well as maps w,2(t, ψ, ε), w,3(t, ψ, ε), 2π–periodic with respect to t, ψ such that for every |ε| < ε3 with Ω(ε) 6∈ Q the transformation

r = r + r2 w,2(t, ψ, ε) + r3 w,3(t, ψ, ε) (3.23)

defined for |r| < r∞ leads

ψ = Ω(ε) + r f ,1(t, ψ, ε) + r2 f ,2(t, ψ, r, ε)

r = r2 g,2(t, ψ, ε) + r3 g,3(t, ψ, r, ε)(3.20)

into a system of the form

ψ = Ω(ε) + r f ,1(t, ψ, ε) + r2 f ,2(t, ψ, r, ε)

˙r = r3(

g,30,0(0, ε) +m0,0(ε))

+ r4 g,4(t, ψ, r, ε),(3.24)

where

g,30,0(r, ε) =1

(2π)2

2π∫

0

2π∫

0

g,3(t, ψ, r, ε) dt dψ

m0,0(ε) =1

(2π)2

2π∫

0

2π∫

0

(

2 g,2(t, ψ, ε)w,2(t, ψ, ε) + ∂ψw,2(t, ψ, ε) f ,1(t, ψ, ε)

)

dt dψ.

(3.25)

PROOF: Since the maps g,2, g,3 are of class C1 and 2π–periodic with respect to t and ψ, we mayconsider the Fourier series

g,2(t, ψ, ε) =∑

k,n∈Z

g,2k,n(ε) ei(kψ+nt)

g,3(t, ψ, r, ε) =∑

k,n∈Z

g,3k,n(r, ε) ei(kψ+nt)

2 g,2(t, ψ, ε)w,2(t, ψ, ε) + ∂ψw,2(t, ψ, ε) f ,1(t, ψ, ε) =

∑

k,n∈Z

mk,n(ε) ei(kψ+nt)

(3.26)

3.4. Averaging the Quadratic and Cubic Term 123

where from proposition 3.3.2 in section 3.3 it follows that g,20,0 = 0. It is evident that the identities (3.25)

define the quantities g,30,0(r, ε), m0,0(ε) respectively. Setting

w,2(t, ψ, ε) = −∑

k,n∈Z

(k,n) 6=(0,0)

g,2k,n(ε)

i(k Ω(ε) + n)ei(kψ+nt)

w,3(t, ψ, ε) = −∑

k,n∈Z

(k,n) 6=(0,0)

g,3k,n(0, ε) +mk,n(ε)

i(k Ω(ε) + n)ei(kψ+nt)

(3.27)

we find from (3.23) and (3.20)

˙r = r + 2 r r w,2(t, ψ, ε) + r2(

∂tw,2(t, ψ, ε) + ∂ψw

,2(t, ψ, ε) ψ)

+3 r2 r w,3(t, ψ, ε) + r3(

∂tw,3(t, ψ, ε) + ∂ψw

,3(t, ψ, ε) ψ)

= r2 g,2(t, ψ, ε) + r3 g,3(t, ψ, r, ε)

+2 r(

r2 g,2(t, ψ, ε) + r3 g,3(t, ψ, r, ε))

w,2(t, ψ, ε)

+3 r2(

r2 g,2(t, ψ, ε) + r3 g,3(t, ψ, r, ε))

w,3(t, ψ, ε)

+r2(

∂tw,2(t, ψ, ε) + ∂ψw

,2(t, ψ, ε) Ω(ε))

+r2 ∂ψw,2(t, ψ, ε)

(

r f ,1(t, ψ, ε) + r2 f ,2(t, ψ, r, ε))

+r3(

∂tw,3(t, ψ, ε) + ∂ψw

,3(t, ψ, ε) Ω(ε))

+r3 ∂ψw,3(t, ψ, ε)

(

r f ,1(t, ψ, ε) + r2 f ,2(t, ψ, r, ε))

. (3.28)

Solving (3.23) with respect to r yields an identity of the form r = r + r2W (t, ψ, r, ε) such that the lastequation may be written in the form

˙r = r2 g,2(t, ψ, ε) + r2(

∂tw,2(t, ψ, ε) + ∂ψw

,2(t, ψ, ε) Ω(ε))

+r3 g,3(t, ψ, 0, ε) + r3(

∂tw,3(t, ψ, ε) + ∂ψw

,3(t, ψ, ε) Ω(ε))

+2 r3 g,2(t, ψ, ε)w,2(t, ψ, ε) + r3 ∂ψw,2(t, ψ, ε) f ,1(t, ψ, ε)

+r3(

g,3(t, ψ, r, ε)− g,3(t, ψ, 0, ε))

+O(r4).

Plugging in the definition of w,2(t, ψ, ε), w,3(t, ψ, ε) respectively yields

˙r = r2 g,2(t, ψ, ε)

−r2(

∑

k,n∈Z

(k,n) 6=(0,0)

i ng,2k,n(ε)

i(k Ω(ε) + n)ei(kψ+nt) +

∑

k,n∈Z

(k,n) 6=(0,0)

i kg,2k,n(ε)

i(k Ω(ε) + n)ei(kψ+nt) Ω(ε)

)

+r3 g,3(t, ψ, 0, ε)

−r3(

∑

k,n∈Z

(k,n) 6=(0,0)

i ng,3k,n(0, ε) +mk,n(ε)

i(k Ω(ε) + n)ei(kψ+nt) +

∑

k,n∈Z

(k,n) 6=(0,0)

i kg,3k,n(0, ε) +mk,n(ε)

i(k Ω(ε) + n)ei(kψ+nt) Ω(ε)

)

+2 r3 g,2(t, ψ, ε)w,2(t, ψ, ε) + r3 ∂ψw,2(t, ψ, ε) f ,1(t, ψ, ε) +O(r4)


= r2 g,2(t, ψ, ε)− r2∑

k,n∈Z

(k,n) 6=(0,0)

g,2k,n(ε) ei(kψ+nt)

+r3 g,3(t, ψ, 0, ε)− r3∑

k,n∈Z

(k,n) 6=(0,0)

(

g,3k,n(0, ε) +mk,n(ε))

ei(kψ+nt)

+r3(

2 g,2(t, ψ, ε)w,2(t, ψ, ε) + ∂ψw,2(t, ψ, ε) f ,1(t, ψ, ε)

)

+O(r4)

= r2 g,20,0(ε) + r3(

g,30,0(0, ε) +m0,0(ε))

+O(r4)

= r3(

g,30,0(0, ε) +m0,0(ε))

+O(r4)

since g,20,0 = 0. This establishes the statement given above.

We complete this chapter on the stability of the invariant set h = 0 by giving a statement on the caseof non–linear (but ”cubic”) stability :

Corollary 3.4.2 The form (3.24) deduced in this section admits the following conclusion :

1. Assume that there exists a positive constant r∞ such that for all 0 ≤ r < r∞ and any |ε| < ε3 theinequality

r <

∣

∣

∣g,30,0(0, ε) +m0,0(ε)

∣

∣

∣

|g,4(t, ψ, r, ε)|

is fulfilled. Then the invariant set r = 0 of (3.24) and hence the invariant set h = 0 of (3.1)is stable if g,30,0(0, ε) +m0,0(ε) < 0 and unstable if g,30,0(0, ε) +m0,0(ε) > 0.

2. Consider the situation of corollary 3.2.3 again where f ,j, g,j are of order O(ε2). By definition(3.27) we see that w,2(t, ψ, ε) = O(ε2) and therefore m0,0(ε) = O(ε4). Choosing ε sufficiently small

the quantity m0,0(ε) is therefore small compared to g,30,0(0, ε). Moreover g,30,0(0, ε) may be written inthe form

g,30,0(0, ε) =1

(2π)2

2π∫

0

2π∫

0

g,3(t, ϕ, 0, ε) dt dϕ+O(ε3) = ε2 g2,30,0 +O(ε3)

where

g2,30,0 =1

(2π)2

2π∫

0

2π∫

0

12∂

2εg,3(t, ϕ, 0, 0) dt dϕ.

In this situation the algebraic (or ”cubic”) stability therefore may be discussed by considering thesign of the corresponding quantity g2,30,0 given by the original vector field (3.1), omitting the explicitcalculation of the transformation w.

Chapter 4

Application to a MiniatureSynchronous Motor

4.1 Introduction

4.1.1 The Physical Model

In this part we will apply the theory of chapters 1–3 to an example which arises in electrical engineering.

i1 U

2

0

iN

S

R

R

L

L

ϑ

m

λ

uC

Bϕ

Figure 4.1: Schematic sketch of the minia-ture synchronous motor considered

Consider a so–called synchronous motor as sketchedin figure 4.1. The type considered here is drivenby alternating current and has a permanent mag-net on the rotor. Two coils situated in an 90–angle are connected parallel to the power supply.In order to produce a rotating magnetic field, oneof the circuits is supplied with a condenser caus-ing a phase shift. It is a typical property of syn-chronous motors that once the rotor is rotating withan angular frequency close to the one of the powersupplied, it stabilizes to this frequency of the cur-rent. However there are different ways to acceler-ate the rotor to this frequency first (pony–motors,induction–cages, . . . ). A special feature of the mo-tor considered here is that there are no such addi-tional mechanisms needed to accelerate the rotor uponstart.

125

126 Chapter 4. Application to a Miniature Synchronous Motor

The simplified physical model of this motor is describedvia the following system of ordinary differential equations:

d2

dτ2ϑ = −λ

J

√

i21 + i22 sin(ϕ) − ˜d

dτϑ− m


dτi1 + λ

d

dτsin(ϑ)


dτi2 + λ

d

dτcos(ϑ) + u

d

dτu = i2/C

(4.1)

where ϕ = ϑ− arg(i2 + i i1). The physical parameters satisfy

U0 ∈ [5, 50] V

L ∈ [0.25, 0.5] V s/A

C ∈ [5 · 10−6, 25 · 10−6] As/V

λ ∈ [0.01, 1.45] V s

R = 100 V/A

ω = 50 · 2π 1/s

J = 5 · 10−8 kgm2.

(4.2)

The term ˜ ddτ ϑ corresponds to a linear damping, the parameter m to an external torque as for instance,

caused by a constant load.

4.1.2 Simplifying Transformations and Assumptions on the Parameters

Using the definition of ϕ we calculate

√

i21 + i22 sin(ϕ) = |i2 + i i1| sin(ϑ− arg(i2 + i i1))

= |i2 + i i1| cos(arg(i2 + i i1)) sin(ϑ)

− |i2 + i i1| sin(arg(i2 + i i1)) cos(ϑ)

= i2 sin(ϑ)− i1 cos(ϑ)

such that system (4.1) reads

d2

dτ2ϑ = −λ

J(i2 sin(ϑ)− i1 cos(ϑ))− ˜

d

dτϑ− m


dτi1 + λ

d

dτsin(ϑ)


dτi2 + λ

d

dτcos(ϑ) + u

d

dτu = i2/C.

(4.3)

If the parameter λ equals zero, system (4.3) decouples and the subsystem of the electrical variables i1,

4.1. Introduction 127

i2, u reads


dτi1


dτi2 + u

d

dτu = i2/C.

(4.4)

This system however admits a unique attractive periodic solution (I1(τ), I2(τ), U(τ)) given by

I1(τ) = −U0 (Lω cos(ω τ )− R sin(ω τ ) )

R2 + ω2 L2

I2(τ) = −U0 ω C(

−cos(ω τ ) + Lω2 cos(ω τ )C − ω sin(ω τ )RC)

R2 C2 ω2 + 1− 2ω2LC + ω4 L2C2

U(τ) = − U0

(

−sin(ω τ ) + L sin(ω τ )ω2 C + ω cos(ω τ )RC)

R2 C2 ω2 + 1− 2ω2LC + ω4 L2 C2.

(4.5)

Due to the linear structure of the electrical subsystem of (4.3) we are able to perform a time–dependentchange of coordinates which transforms it into an autonomous system. We introduce such new coordinatesI1, I2, U as follows:

(τ, i1, i2, u) = (τ, I1, I2, U) + (0, I1(τ), I2(τ), U (τ)) (4.6)

Then system (4.3) transforms to

d2

dτ2ϑ = −λ

J

(

I2(τ) sin(ϑ)− I1(τ) cos(ϑ))

− λ

J

(

I2 sin(ϑ) − I1 cos(ϑ))

− ˜d

dτϑ− m

0 = R I1 + Ld

dτI1 + λ

d

dτsin(ϑ)

0 = R I2 + Ld

dτI2 + λ

d

dτcos(ϑ) + U

d

dτU = I2/C.

(4.7)

Using the explicit forms (4.5) the equation for d2

dτ2ϑ may be simplified for a special choice of parameters.It may be found by (4.5) that

(

I2(τ) sin(ϑ) − I1(τ) cos(ϑ))

=1

2

(

U0 ωC − U0 ω3 C2 L

R2 C2 ω2 + 1− 2ω2LC + ω4L2 C2− U0R

R2 + ω2 L2

)

sin(ϑ+ ω τ )

+1

2

(

U0 Lω

R2 + ω2 L2− U0 ω

2C2 R

R2 C2 ω2 + 1− 2ω2 LC + ω4L2 C2

)

cos(ϑ+ ω τ )

+1

2

(

U0 ω C − U0 ω3C2 L

R2C2 ω2 + 1− 2ω2LC + ω4 L2 C2+

U0R

R2 + ω2 L2

)

sin(ϑ− ω τ )

+1

2

(

U0 Lω

R2 + ω2 L2+

U0 ω2C2 R

R2 C2 ω2 + 1− 2ω2 LC + ω4L2 C2

)

cos(ϑ− ω τ ).


Taking into account that all parameters are positive, the coefficient of cos(ϑ − ωτ) is non–zero. On theother hand, the coefficients of sin(ϑ+ ωτ) and of cos(ϑ+ ωτ) vanish if (and only if)

L =R

ωC =

1

2ωR, (4.8)

which may be fulfilled by the parameters of the system (4.1) considered. Hence in what follows we willassume that (4.8) holds. For this case, (4.7) simplifies to

(

I2(τ) sin(ϑ)− I1(τ) cos(ϑ))

=U0

2R(sin(ϑ− ω τ ) + cos(ϑ− ω τ ))

=U0√2R

sin (ϑ− ω τ + π/4)

such that (4.7) can be written as

1

ω2

d2

dτ2ϑ = − λU0√

2J Rω2sin(ϑ− ωτ + π/4)− λ

J ω2

(

I2 sin(ϑ)− I1 cos(ϑ))

− ˜

ω2

d

dτϑ− m

ω2

1

ω

d

dτ

R I1 + λω sin(ϑ)

U0= −R I1

U0

1

ω

d

dτ

R I2 + λω cos(ϑ)

U0= −R I2 + U

U0

1

ω

d

dτ

U

U0= 2

R I2U0

.

(4.9)

This representation (4.9) motivates a further change of coordinates in the (ϑ, I1, I2, U)–space. For α > 0fixed, we introduce the vector η = (η1, η2, η3)

T ∈ R3 by

η1 := αR I1 + λω sin(ϑ)

U0

η2 := αR I2 + λω cos(ϑ)

U0

η3 := αU

2U0,

(4.10)

and rescale the time variable as follows :

t = ω τ − π/4 (4.11)

One immediately calculates the transformed system of (4.9) in the new coordinates, which unlike (4.9)contains no terms d

dτ sin(ϑ),ddτ cos(ϑ) anymore:

d2

dt2ϑ = − λU0√

2 J Rω2sin(ϑ− t) +

1

α

λU0

J Rω2(η1 cos(ϑ)− η2 sin(ϑ))−

˜

ω

d

dtϑ− m

ω2

d

dtη1 = −η1 + α

λω

U0sin(ϑ)

d

dtη2 = −η2 − 2 η3 + α

λω

U0cos(ϑ)

d

dtη3 = η2 − α

λω

U0cos(ϑ).

(4.12)


4.1.3 Transformation into the Form as Discussed in Chapter 1

For a fixed magnetic dipol of the rotor one expects, that if the mass and therefore the inertia of rotationJ of the rotor is increased, in order to regain a stable behaviour, the voltage U0 has to be increased aswell. We simulate this fact by a first, linear approximation of the form

U0 = a J

for a suitable constant a > 0. In order to simplify the notation in what follows, we introduce someabbreviations, fix the parameter α and perform a time–dependent shift of the ϑ–coordinate :

(a

2

)2

:=λU0√2J Rω2

=λ a√2Rω2

ε :=λ√J Rω

=λ√a√

U0Rω

α :=U0√J Rω3

= ε−1√2(a

2

)2

:=˜

ε2 ωm :=

m

ε2 ω2

q := ϑ− t p :=d

dtϑ− 1 (4.13)

Remark 4.1.1 Choosing ε as a perturbation parameter may be understood as follows :

If we increase the voltage U0 of the circuit, the magnetic field B will grow as well. Thus the forces actingon the magnetic dipol of the rotor will be large. In order to prevent the rotor from overreaction, we haveassumed that the inertia of rotation increases together with the voltage U0. The accelerations of the rotor,caused by the magnetic field, are then expected to be qualitatively invariant.

However, as the moment of magnetic dipol λ is fixed the influence of the rotating magnet on the coilsremains constant as U0 increases. Hence the voltage of induction from the coils back to the circuit remainssmall while U0 increases.

Therefore the limit ε→ 0 may be interpreted as taking away the effect of the rotating permanent magneton the circuit and considering the influence of the magnetic field (caused by the current in the circuit) onthe rotor solely.

Applying (4.1.3) on (4.12) yields a system of the form (1.1) considered in chapter 1, namely

q = p

p = −(a

2

)2

sin(q) + ε (η1 cos(q + t)− η2 sin(q + t))− ε2 p− ε2 (m+ )

η1 = −η1 + ε sin(q + t)

η2 = −η2 − 2 η3 + ε cos(q + t)

η3 = η2 − ε cos(q + t).

(4.14)

Hence we are in the position to discuss the model under consideration using the theory derived in theprevious chapters.

Note that the ranges of the various parameters as listed in (4.2) imply a = 2√

λU0√2 J Rω3

∈ [0.54, 20.38].


Before entering the discussion of (4.14) we prove the following statements on the relation between theoriginal ”physical” system (4.1) and the transformed system (4.14):

Lemma 4.1.2

a) Consider the solution (ϑ, i1, i2, u)(τ) of system (4.1) with initial values

ϑ(τ0) = ϑ0,d

dτϑ(τ0) = 0

i1(τ0) = i2(τ0) = u(τ0) = 0

at time τ0 = π4ω . Transforming this solution into (q, p, η)–coordinates, this yields the uniquely

determined solution of (4.14) with initial condition

q(0) = ϑ0, p(0) = −1

η1(0) = ε sin(ϑ0), η2(0) = ε cos(ϑ0)− ε−1(a

2

)2

, η3(0) = 0

at t = 0.

b) Assume that there exists a solution (q, p, η)(t) of (4.14) with

|p(t)| ≤ p∞ <∞ ∀t ≥ t0.

Then the corresponding solution of (4.1) satisfies

∣

∣

∣

∣

d

dτϑ(τ) − ω

∣

∣

∣

∣

≤ p∞ω

∀τ ≥ t0 +π4

ω.

Hence every solution of (4.14) with p(t) bounded (and small) is equivalent to a rotation of thesynchronous motor with the mean frequency ω of the power supply.

PROOF: Let us first symplify the expressions given in (4.5) using (4.8) and (4.11):

I1(τ) = − U0

2R(cos(ω τ)− sin(ω τ)) =

U0√2R

sin(t)

I2(τ) = − U0

2R(− cos(ω τ) − sin(ω τ)) =

U0√2R

cos(t)

U(τ) = −U0 (cos(ω τ)− sin(ω τ)) = U0

√2 sin(t).

Summarizing the transformations given in (4.6) and (4.10) we then find

i1 =1

R

(

U0

αη1 − λω sin(ϑ)

)

+U0√2R

sin(t)

i2 =1

R

(

U0

αη2 − λω cos(ϑ)

)

+U0√2R

cos(t)

u =2U0

αη3 + U0

√2 sin(t)


By (4.11) we see that τ = π4ω ⇔ t = 0, such that by (4.1.3)

ϑ(π

4ω) = ϑ0,

d

dτϑ(

π

4ω) = 0 ⇐⇒ q(0) = ϑ0, p(0) = −1.

Moreover we see that αU0λω = λω√

J Rω3= ε which implies

i1(π

4ω) = 0 ⇔ η1(0) =

α

U0λω sin(ϑ0) = ε sin(ϑ0)

i2(π

4ω) = 0 ⇔ η2(0) =

α

U0

(

λω cos(ϑ0)−U0√2

)

= ε cos(ϑ0)− ε−1(a

2

)2

u(π

4ω) = 0 ⇔ η3(0) = 0,

which proves a). The statement given in b) is a simple consequence of (4.11) and (4.1.3), since

|p| =∣

∣

∣

∣

d

dtϑ(τ) − 1

∣

∣

∣

∣

=

∣

∣

∣

∣

d

dτϑ(τ) − ω

∣

∣

∣

∣

/ω ≤ p∞ω.

The aim of this chapter is to show that for a large set of initial values ϑ0 ∈ [0, 2π] as considered in a), theasymptotic behaviour of the solutions of the physical system (4.1) corresponds to a uniform movementof the rotor as described in b). This will be proved by applying the theory derived in the preceedingchapters.

Recall that sufficient information on the asymptotic behaviour of (4.14) may be found if the coeffi-cient maps gjk,n (as considered in chapter 2) and g,10,0 (as in chapter 3) are known. We will apply the

transformations (introduced in chapter one) which lead to the formulae for g2k,n in the case of system(4.14). However, since these preparations include intensive algebraic manipulations, we will make use ofthe Maple [15] software package for symbolic algebraic computations. The same software will be usedeventually to carry out the numerical calculations necessary to approximate the values g2k,n(h).

Considering (4.14), we see that this system is of the general form (1.1) considered in chapter 1 whered = 3 and the matrix A, the Hamiltonian H and the maps F and G are as follows:

A =

−1 0 00 −1 −20 1 0

H(q, p) =p2

2+(a

2

)2

(1− cos(q))

F (q, p, η, t, ε) = ε

(

0η1 cos(q + t)− η2 sin(q + t)

)

− ε2(

0 p+ (m+ )

)

(4.15)

G(q, p, t, ε) = ε

sin(q + t)cos(q + t)− cos(q + t)

.


4.1.4 Proof of the General Assumptions GA1

It is evident that the functions F and G are of class Cω, 2π–periodic and fulfill F (q, p, t, η, 0) = 0,G(q, p, t, 0) = 0 for all q, p, t, η. As a next step we will establish the properties assumed in the GeneralAssumption of section 1.1.2.

1. The unperturbed Hamiltonian system ddt (q, p) = J∇H(q, p) corresponds to the pendulum equation

q = p

p = −(a

2

)2

sin(q).(4.16)

(a) Using the definition of the Hamiltonian H we calculate√

∂2qH(0, 0) ∂2pH(0, 0) = a2 . If we

restrict ourselves to parameters a ∈ R∗+ \ 2Z then all assumptions made on the function H are

satisfied.

(b) For any p0 ∈ (−a, a) the solution (q, p)(t; 0, p0) of (4.16) with initial value (0, p0) at time t = 0corresponds to an oscillatory (or constant) solution of the pendulum equation and may beexpressed by

sin(12q(t; 0, p0)) =p0a

sn(a

2t;p0a

)

p(t; 0, p0) = p0 cn(a

2t;p0a

)

.(4.17)

where sn and cn are the Jacobian Elliptic Functions (cf. formula 6.17 in [7]). The frequencyof these solutions is given by

Ω(p0) =a

2

2π

4K(

p0a

) > 0 (4.18)

where K is the Elliptic Integral of the First Kind.

(c) As K is sufficiently regular, we see that choosing any Jr ∈ (0, a) (and setting Jl := −Jr), thelimit dk

dp0kΩ(p0) for p0 → Jl,Jr exists for all k ∈ N. We will denote the corresponding central

domain of action angle coordinates by JC . Since K(0) = π2 we eventually find

Ω(0) := limp0→0+

Ω(p0) =a

2> 0

and the formulae (710.00), (111.02) in [2] imply

ddp0

Ω(0) = 0.

2. The eigenvalues of the matrix A defined in (4.1.3) are given by − 12 + i

√72 ,− 1

2 − i√72 and −1 such

that the real parts of these eigenvalues are bounded by − 12 and A is diagonalizable. This establishes

the assumption GA 1.2.

3. Set F j := 0 (j = 3, 4) and Gj := 0 (j = 2, 3, 4) and let

F 1(q, p, η, t) :=

(

0η1 cos(q + t)− η2 sin(q + t)

)

, F 2(q, p, η, t) :=

(

0− p− (m+ )

)

G1(q, p, t) :=

sin(q + t)cos(q + t)− cos(q + t)

.

(4.19)


Then F and G may be represented in the form (1.3) assumed in GA 1.3. These functions F 1 andG1 defined above are 2π–periodic with respect to t and may be expanded to a Fourierpolynomialof degree N = 1. To gain a representation as in (1.4), we define the quantities

M :=

[

0 0 012

i2 0

]

v :=

− i2

12

− 12

. (4.20)

Setting

F 11 (q, p, η) := eiqMη F 1

−1(q, p, η) := e−iqMη

F 20 (q, p, η) :=

(

0− p− (m+ )

)

G11(q, p) := eiqv G1

−1(q, p) := e−iqv

F jn(q, p, η) := 0 Gjn(q, p) := 0 else,

(4.21)

we find

F 1(q, p, η, t) = F 11 (q, p, η) e

it + F 1−1(q, p, η) e

−it

F 2(q, p, η, t) = F 20 (q, p, η)

G1(q, p, t) = G11(q, p) e

it +G1−1(q, p) e

−it.

(4.22)

4. The existence of a map P as assumed in 1.97 a–1.97d is evident. Since this map has been introducedfor technical reasons only and does not influence any qualitative statements given, we will need noparticular choice and may therefore consider any function P which satisfies 1.97 a–1.97d.

5. In view of definition (4.1.3) we see that F is affine with respect to the vector η. Thus GA 1.4 holdsas well.

This proves all assumptions made in GA1 for the system considered in (4.14). The assumptions made inGA2 are established at once. In particular the properties assumed for the map ω are shown by applyinggeneral results on the Elliptic Integral of the First Kind. Hence the theory derived in the preceedingchapters may be applied on the model of the miniature synchronous motor considered here.


For the choice Jl ' a, Jr = ∞, JU := (Jl,Jr) defining the upper domain or Jl = −∞, Jr / a,JL := (Jl,Jr) defining the lower domain, it may be shown that these assumption are fulfilled as well.The corresponding discussion then focuses on the regions of rotatory solutions outside the ”eye–shapedregion” formed by the separatrices of the unperturbed pendulum in the (q, p)–space (cf. figure 4.2) . Theadditional coordinates η will however, take all values in R3 indepently of the choice of J .

η

q

-p

L J

JL

C

L

L

x

UJ

3

x3

R

R

x3

R

Figure 4.2: The three domains admitting action angle coordinates (η simplified to one dimension)

The process to derive the formulae necessary to execute the calculations of the values g2k,n(h) for JU andJL is analogous to the one given here in the case JC and therefore omitted. In section (4.5) we will givethe results for the upper domain and the lower domain as well.

The expression p0a in (4.17) corresponds to the modulus of the elliptic functions with respect to oscillatory

solutions, i.e. in the central domain JC . For the deduction of the formulae corresponding to the upperand lower domains JU and Jl the expression in (4.17) must be adapted to the rotary solutions. Themodulus for this case is given by ± a

p0.

4.2. Preliminary Discussion and Numerical Simulations 135

4.2 Preliminary Discussion and Numerical Simulations

In order to gain a first overview on the qualitative behaviour of (4.14) and its dependence on the parame-ters, we present a list of results found through numerical simulations of system (4.14). These simulationswere carried out using the dstool–software package [3].

4.2.1 The Role of the Coupling and the Parameters , m

Let us rewrite system (4.14) in the slightly more general form

q = p

p = −(a

2

)2

sin(q) + µ ε (η1 cos(q + t)− η2 sin(q + t))− ε2 (p+ 1)− ε2m

η1 = −η1 + µ ε sin(q + t)

η2 = −η2 − 2 η3 + µ ε cos(q + t)

η3 = η2 − µ ε cos(q + t)

(4.23)

i.e. by introducing an additional parameter µ which enables us to control the amount of coupling be-tween the two subsytems of the (q, p) and the η–coordinates. Considering the (q, p)–plane of the phasespace solely, system (4.23) then may be studied for special choices of the parameters µ, and m. Thecorresponding phase portraits are depicted in figure 4.3. The parameter a is fixed to a = 2.33.

ρ

m

µ

Figure 4.3: Dependence of the phase portrait of (4.23) on the parameters


1. µ = 0, = 0, m = 0 :

In this case the system considered is given by

q = p

p = −(a

2

)2

sin(q)

hence the standard mathematical pendulum equation. This system is Hamiltonian and the separa-trices cross the p–axis at p = ±a.

2. µ = 0, = 0, m 6= 0 :

The system considered is given by

q = p

p = −(a

2

)2

sin(q)− ε2m

corresponding to the equation of a mathematical pendulum with a small external torque. Thissystem is Hamiltonian as well.

3. µ = 0, 6= 0, m = 0 :

Then

q = p

p = −(a

2

)2

sin(q)− ε2 (p+ 1)

corresponding to the equation of a mathematical pendulum with small constant damping andexternal torque. This system is dissipative for > 0.

4. µ 6= 0, = 0, m = 0 :

In this last case we have

q = p

p = −(a

2

)2

sin(q) + µ ε (η1 cos(q + t)− η2 sin(q + t))

η1 = −η1 + µ ε sin(q + t)

η2 = −η2 − 2 η3 + µ ε cos(q + t)

η3 = η2 − µ ε cos(q + t)

corresponding to the mathematical pendulum with a weak coupling to the η system. This systemis non–autonomous and numerical simulation yields a phase portrait which suggest an attractiveperiodic solution close to the origin. (Note that q = p = 0, η = 0 does not solve the system).


4.2.2 The Role of the Parameter a

In order to obtain a first overwiew on the qualitative behaviour of (4.14), (4.23) respectively and inparticular the effect that the coupling of the two subsystems takes, we set = 0, m = 0 (and µ = 1).For the parameter a we consider a = 0.54 and a = 20.38, i.e. the lower and upper bound of the domainconsidered in (4.14). Since a = 4.0 corresponds to a resonant case (where GA 1.1a is not satisfied, seesection 4.1.4), we consider the value a = 4.1 in addition.

For each choice of the parameter a we depict the following plots: For the solutions (q, p)(t) of (4.14)considered, the transformation (1.15) into the periodic solution (q, p, η)(t, ε) and hence into the (Q, P )–coordinates is approximated numerically. Then the graph of the function t 7→ H((Q, P )(t)) is shown in afirst figure. The next figure shows the projections of the phase portrait onto the (q, p)–plane. The orbitsof a few solutions are shown near the eye–shaped region. Then the same trajectories are shown with alarge zoom out on the q–axis. This makes it possible to track solutions more globally and to observe thelong time behaviour.

For a = 0.54 the final figure shows a view on the (p, η2)–plane in order to illustrate the behaviour in theη–space and to demonstrate how solutions approach η ≈ 0. This figure is very similar for a = 4.1 anda = 20.38 and therefore omitted for these parameters.

Note that due to extreme zoom out, the trajectories plotted may be too tight to be distinguished andseemingly fill an entire area.

a = 0.54, = 0, m = 0, (ε = 0.05)

real–world initial values: We first simulate the behaviour of the synchronous motor when switched on.The corresponding initial values are refered to as the real–world initial values. For a = 0.54 we chooseeight equidistant values for ϑ(0) ∈ [−π, π], d

dτ ϑ(0) = 0 and transform them in accordance to lemma 4.1.2.The corresponding trajectories then are plotted in black.As visible in figure 4.4 they limit towards p = −1 which corresponds to d

dtϑ = 0. The η–componentsof these solutions approach a small neighbourhood of η = 0 after some transient behaviour. The corre-sponding trajectories are shown in figure 4.5. This plot illustrates the existence of an attractive invariantmanifold close to η = 0, as expected by the results found in chapter 1.We conclude that the power circuit enters some periodic behaviour (cf. transformations (4.6), (4.10)) butthe rotor does not start rotating when the motor is switched on. In figure 4.6 we see that H(Q, P ) isstrictly bounded away from zero. As H = 0 is equivalent to (Q, P ) = 0 (modulo 2π in Q) this showsagain that the solutions with real–world initial values remain O(1) away from the periodic solution of(4.14), i.e. the synchronous rotary behaviour.

reduced system: Secondly we consider initial values with η = 0, i.e. close to the attractive invariantmanifold. This yields a more extensive view of the reduced system. More precisely we show trajectoriescorresponding to q(0) = 0, η(0) = 0 and p(0) ∈ [−2, 2]. These trajectories are plotted in grey. Thefollowing result may be seen in figure 4.7 and figure 4.8 best:For p(0) / −0.54 the orbits limit in p = −1. For −0.54 ' p(0) / 0.54 the orbits are caught by theperiodic solution close to the origin (refer also to figure 4.6 where H((Q, P )(t)) → 0). For p(0) ' 0.54some solutions are caught by this periodic solution as well, other solutions pass the q–axis and limit inp = −1.


t

p

Figure 4.4: t ∈ [0, 2000], p ∈ [−2, 2]

η

p

2

Figure 4.5: p ∈ [−2, 2], η2 ∈ [−2, 2]


H(Q,P)

t

Figure 4.6: t ∈ [0, 2000], H(Q, P ) ∈ [0, 1]

q

p

p=-1

Figure 4.7: q ∈ [−4, 4], p ∈ [−2, 2]

q

p

Figure 4.8: q ∈ [−500, 500], p ∈ [−2, 2]


a = 4.1, = 0, m = 0, (ε = 0.05)

real–world initial values: We consider solutions starting with three different values for ϑ(0). In orderto visualize the long time behaviour we have plotted the corresponding trajectories with time–dependentcolour. For t = 0 the orbits are light grey and become black as t → ∞. Two orbits correspondingto ϑ(0) = −2 and ϑ(0) = −1.1 tend towards the periodic solution close to the origin within the time0 ≤ t ≤ 2000 integrated numerically. A third orbit corresponding to ϑ(0) = −1.04 is plotted last andcovers the previous two trajectories. Due to some capture in a resonance it is caught in an attractor insidethe eye–shaped region (cf. section 4.5). This is seen in figure 4.11 as the orbit covers itself increasinglythe darker it becomes and the black part of the trajectory therefore corresponds to the ω–limit set. Thisω–limit set inside the attractor has the shape of a circle and intersects the p–axis at p ≈ 1.2.

reduced system: Trajectories corresponding to initial values with q(0) = 0, η(0) = 0 and p(0) ∈ [−6, 6]are coloured in grey. All these orbits are attracted by the eye–shaped region eventually (figure 4.12).This is seen in figure 4.10, too, as the energies H(Q, P ) become less than 8 as t → ∞ and the set

Q, P∣

∣H(Q, P ) ≤ 8

lies within the eye–shaped region. The plot of H((Q, P )(t)) versus t illustratesthe capture in the resonance as well: In figure 4.9 this function is evaluated for 20 solutions with initialvalues p(0) ∈ [1, 3]. For three of these solutions the energy approaches the value ≈ 0.72 = H(0, 1.2)indicating a capture in the resonance. (The same is visible for the solution with the real–world initialvalue ϑ(0) = −1.04 in figure 4.10.)

H(Q,P)

t

Figure 4.9: t ∈ [0, 6000], H(Q, P ) ∈ [0, 2]


H(Q,P)

t

Figure 4.10: t ∈ [0, 3000], H(Q, P ) ∈ [0, 10]

q

p

p=-1

Figure 4.11: q ∈ [−4, 4], p ∈ [−6, 6]

q

p

Figure 4.12: q ∈ [−3000, 3000], p ∈ [−6, 6]


In order to illustrate the ε–dependence of the amount of solutions being captured in the resonancepresumed, we plot q(t) over the values

σ = δ t+ q(0)

for ten equidistant values q(0) ∈ [−3,−0.5] and three different choices for ε. The result is shown infigure 4.13 where we have used different shades of grey for colouring and different values for δ accordingto the following choices of ε:

black ε = 0.4, δ = 3 · 10−5 ⇒ 5 orbits are caught in the resonance

dark grey ε = 0.2, δ = 2 · 10−5 (covers previous plot) ⇒ 3 orbits are caught in the resonance

light grey ε = 0.1, δ = 1 · 10−5 (covers previous plots) ⇒ 2 orbits are caught in the resonance.

σ

q

Figure 4.13: σ ∈ [−3,−0.5], q ∈ [−4, 4]

Hence the number of solutions captured in the resonance decreases as ε → 0. Note that the size ofthe attracting periodic solution q(t, ε) decreases as ε → 0 while the maximal q–values of the solutionscaptured remains constant in the main as this corresponds to the ε–independent position of the resonance.

a = 20.38, = 0, m = 0, (ε = 0.1)

The colouring of the plots is processed as for a = 4.1, i.e. time–dependent for the real–world initial valuesand grey for trajectories close to the reduced system.

real–world initial values: Since the solutions corresponding to this set of initial values start on the linep = −1 most trajectories start within the eye–shaped region and are caught by the periodic solution nearthe origin. The orbits starting outside the eye–shaped region perform a few shifts in the q–coordinate,then drift into the eye–shaped region as well (see figure 4.16). No capture into resonance is found. Asthe numerical integration is performed over the larger time scale t ∈ [0, 4000] and the orbits are coloureddarker with increasing time again, we see in figure 4.15 that the solutions tend slower towards the periodicsolution at the origin, i.e. the periodic solution is less attractive (compared to the preceeding case a = 4.1).

reduced system: The solutions with initial values corresponding to q(0) = 0, η(0) = 0 and p(0) ∈[−30, 30] are attracted by the eye–shaped region, too.


H(Q,P)

t

Figure 4.14: t ∈ [0, 4000], H(Q, P ) ∈ [0, 400]

q

p

p=-1

Figure 4.15: q ∈ [−4, 4], p ∈ [−30, 30]

q

p

Figure 4.16: q ∈ [−400, 400], p ∈ [−30, 30]


4.3 Explicit Formulae for the Reduced System, Following Chap-

ter 1

The purpose of this section is to carry out the programme of chapter 1 in the specific case of (4.14). Basedon the general formulae for the system after each transformation (transformations were performed intothe periodic solution, the strongly stable manifold, into action angle coordinates and on the attractiveinvariant manifold) we eventually will obtain explicit representations of the maps gjk,n as in lemma 1.6.9.

Although the formulae given in chapter 1 enable the computation of gjk,n for j = 2 and j = 3, we will

deal with the ε2–terms only. This will be sufficient to discuss the model under investigation.

4.3.1 The Transformation into the Periodic Solution

Recall the results given in proposition 1.2.4 : For the map F as in (4.1.3) it follows from (4.19)–(4.22)that (1.18) takes the form

F 1(Q, P ,H, t) = eiQM H eit + e−iQM H e−it (4.24)

F 2(Q, P ,H, t) = −(

0 P + (m+ )

)

+∆(0, Q, P )

(

0(m+ )

)

+[

e−iQM −∆(−2, Q, P )M]

α1,1−1,2 e

−i2t

+[

e−iQM −∆(0, Q, P )M]

α1,11,2

+[

eiQM −∆(0, Q, P )M]

α1,1−1,2

+[

eiQM −∆(2, Q, P )M]

α1,11,2 e

i2t (4.25)

G1(Q, P , t) =(

eiQ − 1)

v eit +(

e−iQ − 1)

v e−it (4.26)

where

α1,11,2 = [i IR3 −A]−1 v α1,1

−1,2 = [−i IR3 −A]−1 v.

Defining

B0 :=

[

0 10 0

]

B1 := B−1 :=

[

0 0

−a2

8 0

]

. (4.27)

we write JD2H(Q, P ) =

[

0 1

−(

a2

)2cos(Q) 0

]

in the form

JD2H(Q, P ) = B0 + eiQB1 + e−iQB−1

which implies

∆(n, Q, P ) =[

i n IC 2 − JD2H(Q, P )] [

i n IC 2 − JD2H(0, 0)]−1

= ∆0(n) + ∆1(n) eiQ +∆−1(n) e

−iQ(4.28)

4.3. Explicit Formulae for the Reduced System, Following Chapter 1 145

where

∆0(n) :=[

i n IC 2 −B0] [i n IC 2 −B0 −B1 −B−1]−1

∆1(n) := −B1 [i n IC 2 −B0 −B1 −B−1]−1 ∆−1(n) := −B−1 [i n IC 2 −B0 −B1 −B−1]

−1 .

Using these abbreviations we rewrite (4.25) as

F 2(Q, P ,H, t) = −(

0P

)

− (m+ )(

IC 2 −∆(0, Q, P ))

(

01

)

+[

z00 + z+0 eiQ + z−0 e

−iQ]

+[

z02 + z+2 eiQ + z−2 e

−iQ] ei2t +[

z0−2 + z+−2 eiQ + z−−2 e

−iQ] e−i2t(4.29)

with

z02 = −∆0(2)M α1,11,2, z0−2 = −∆0(−2)M α1,1

−1,2

z+2 = (IC 2 −∆1(2)) M α1,11,2, z+−2 = −∆1(−2)M α1,1

−1,2

z−2 = −∆−1(2)M α1,11,2, z−−2 = (IC 2 −∆−1(−2)) M α1,1

−1,2

z00 = −∆0(0)M α1,11,2 −∆0(0)M α1,1

−1,2

z+0 = (IC 2 −∆1(0)) M α1,1−1,2 −∆1(0)M α1,1

1,2

z−0 = (IC 2 −∆−1(0)) M α1,11,2 −∆−1(0)M α1,1

−1,2.

Computations in Maple [15] yield the following results :

α1,11,2 :=

[

− 1

4− 1

4i

3

4− 1

4i − 1

4− 1

4i

]T

α1,1−1,2 :=

[

− 1

4+

1

4i

3

4+

1

4i − 1

4+

1

4i

]T

(

IC 2 −∆(0, Q, P ))

(

01

)

=

(

01− cos(Q)

)

z02 :=

[

0 4i

−16 + a2

]T

z0−2 :=

[

0 − 4i

−16 + a2

]T

z+2 :=

[

01

4i− 1

8

i a2

−16 + a2

]T

z−−2 :=

[

0 − 1

4i+

1

8

i a2

−16 + a2

]T

z+−2 :=

[

01

8

i a2

−16 + a2

]T

z−2 :=

[

0 − 1

8

i a2

−16 + a2

]T


z00 := [ 0 0 ]T

z+0 :=

[

01

2i

]T

z−0 :=

[

0 − 1

2i

]T

4.3.2 The Transformation into the Strongly Stable Manifold

In this section we refer to the statement given in proposition 1.4.9. In accordance with (4.24)–(4.30) weapply (1.90) (note that F 2 is evaluated for H = 0):

F 1(Q,P,H, t) =( (

eiQ − 1)

B1 +(

e−iQ − 1)

B−1

)

V1(t)H

+(

eiQ − 1)

M H eit +(

e−iQ − 1)

M H e−it(4.30)

F 2(Q,P, 0, t) = −(

0P

)

− (m+ )

(

01− cos(Q)

)

+[

z00 + z+0 eiQ + z−0 e

−iQ]

+[

z02 + z+2 eiQ + z−2 e

−iQ] ei2t +[

z0−2 + z+−2 eiQ + z−−2 e

−iQ] e−i2t

− V1(t)[

(

eiQ − 1)

v eit +(

e−iQ − 1)

v e−it]

.

(4.31)

From (1.91) together with (4.26) we deduce

G1(Q,P,H, t) =(

eiQ − 1)

v eit +(

e−iQ − 1)

v e−it. (4.32)

It remains to compute the linear map V1(t). We therefore consider the decomposition

esA =

3∑

j=1

esλj TA,λj e−sB =

2∑

k=1

e−sωk TB,ωk

implied by (1.82), where

λ1 := −1, λ2 := − 12 + i

√72 , λ3 := − 1

2 − i√72 ω1 := i

a

2, ω2 := −i a

2

and

TA,λ1 :=

1 0 00 0 00 0 0

TA,λ2 :=

0 0 0

01

14i√7 +

1

2

2

7i√7

0 − 1

7i√7 − 1

14i√7 +

1

2

TA,λ3 :=

0 0 0

0 − 1

14i√7 +

1

2− 2

7i√7

01

7i√7

1

14i√7 +

1

2

TB,ω1 :=

1

2− i

a

1

4i a

1

2

TB,ω2 :=

1

2

i

a

− 1

4i a

1

2


Taking into account (4.24) which implies ∂HF1(0, 0, 0, t) = M eit +M e−it, hence ∂HF 1

1 (0, 0, 0, t) = M ,∂HF

1−1(0, 0, 0, t) =M , we determine V1(t) in agreement with (1.83), (1.84):

V1(t) = eitV11 + e−it V1

−1 (4.33)

where

V11 =

3∑

j=1

2∑

k=1

(i− ωk + λj)−1 TB,ωkM TA,λj

V1−1 =

3∑

j=1

2∑

k=1

(−i− ωk + λj)−1 TB,ωkM TA,λj .

Evaluation of these sums using Maple [15] yields

V11 :=

2

−8 i+ a2−24 i+ 2 i a2

−32 i− 20 a2 − 8 i a2 + a4−32− 16 i

−32 i− 20 a2 − 8 i a2 + a4

−2 + 2 i

−8 i+ a28 i− 2 i a2 − 8− 2 a2

−32 i− 20 a2 − 8 i a2 + a4−4 i a2 + 16 i+ 16

−32 i− 20 a2 − 8 i a2 + a4

V1−1 :=

2

8 i+ a224 i− 2 i a2

32 i− 20 a2 + 8 i a2 + a4−32 + 16 i

32 i− 20 a2 + 8 i a2 + a4

−2− 2 i

8 i+ a2−8 i+ 2 i a2 − 8− 2 a2

32 i− 20 a2 + 8 i a2 + a44 i a2 − 16 i+ 16

32 i− 20 a2 + 8 i a2 + a4

.

Plugging (4.33) into (4.30), (4.31) we obtain a representation for F 1(Q,P,H, t) and F 2(Q,P,H, t) whichis more convenient for the further process:

F 1(Q,P,H, t) =[

ζ+1(

eiQ − 1)

+ ζ−1(

e−iQ − 1) ]

H eit

+[

ζ+−1

(

eiQ − 1)

+ ζ−−1

(

e−iQ − 1) ]

H e−it(4.34)

with

ζ+1 = B1 V11 +M ζ−−1 = B−1 V1

−1 +M

ζ+−1 = B1 V1−1 ζ−1 = B−1 V1

1

and

F 2(Q,P, 0, t) = −(

0P

)

− (m+ )

(

01− cos(Q)

)

+[

z00 + z+0 eiQ + z−0 e

−iQ]

+[

z02 + z+2 eiQ + z−2 e

−iQ] ei2t +[

z0−2 + z+−2 eiQ + z−−2 e

−iQ] e−i2t(4.35)

where

z00 = z00 + V11 v + V1

−1 v

z+0 = z+0 − V1−1 v

z−0 = z−0 − V11 v

z02 = z02 + V11 v z0−2 = z0−2 + V1

−1 v

z+2 = z+2 − V11 v z−−2 = z−−2 − V1

−1 v

z−2 = z−2 z+−2 = z+−2.


The corresponding explicit formulae are as follows :

ζ+1 :=

0 0 0

a2 − 16 i

−32 i+ 4 a2−28 i a2 + i a4 + 64 + 16 a2

−128 i− 80 a2 − 32 i a2 + 4 a44 a2 + 2 i a2

−32 i− 20 a2 − 8 i a2 + a4

ζ−−1 :=

0 0 0

a2 + 16 i

32 i+ 4 a228 i a2 − i a4 + 64 + 16 a2

128 i− 80 a2 + 32 i a2 + 4 a44 a2 − 2 i a2

32 i− 20 a2 + 8 i a2 + a4

ζ+−1 :=

0 0 0

− a2

32 i+ 4 a2−12 i a2 + i a4

128 i− 80 a2 + 32 i a2 + 4 a44 a2 − 2 i a2

32 i− 20 a2 + 8 i a2 + a4

ζ−1 :=

0 0 0

a2

−32 i+ 4 a212 i a2 − i a4

−128 i− 80 a2 − 32 i a2 + 4 a44 a2 + 2 i a2

−32 i− 20 a2 − 8 i a2 + a4

z00 :=[

−49152 a2−6144 a4

32768 a2−40 a10+65536+a12+30720 a4−2048 a6+528 a832768+36864 a2−4 a10−1216 a6+6656 a4+96 a8

32768 a2−40 a10+65536+a12+30720 a4−2048 a6+528 a8

]

z+0 :=[

−24 i a2−32 a2+2 i a4−128 i−256−128 i a2−20 a4−64 a2+16 i a4+a6

128+112 a2−20 i a4−12 a4+i a6

−512−256 i a2−40 a4−128 a2+32 i a4+2 a6

]

z−0 :=[

24 i a2−32 a2−2 i a4+128 i−256+128 i a2−20 a4−64 a2−16 i a4+a6

128+112 a2+20 i a4−12 a4−i a6−512+256 i a2−40 a4−128 a2−32 i a4+2 a6

]

z02 :=[

16 i a2+16 a2−64−128 i−256+128 i a2−20 a4−64 a2−16 i a4+a6

−256 a2−136 i a4+192 i a2−2048 i+48 a4+6 i a6

4096−2048 i a2+768 a2+384 i a4+256 a4−36 a6−16 i a6+a8

]

z0−2 :=[

−16 i a2+16 a2−64+128 i−256−128 i a2−20 a4−64 a2+16 i a4+a6

−256 a2+136 i a4−192 i a2+2048 i+48 a4−6 i a6

4096+2048 i a2+768 a2−384 i a4+256 a4−36 a6+16 i a6+a8

]

z+2 :=[

64−16 i a2+128 i−16 a2

−256+128 i a2−20 a4−64 a2−16 i a4+a6i a8+2048 a2+1024 i a4−1792 i a2+16384 i−512 a4+16 a6−68 i a6

32768−16384 i a2+6144 a2+3072 i a4+2048 a4−288 a6−128 i a6+8 a8

]

z−−2 :=[

16 i a2−16 a2+64−128 i−256−128 i a2−20 a4−64 a2+16 i a4+a6

−i a8+2048 a2−1024 i a4+1792 i a2−16384 i−512 a4+16 a6+68 i a6

32768+16384 i a2+6144 a2−3072 i a4+2048 a4−288 a6+128 i a6+8 a8

]

z−2 :=

[

0 − i a2

−128 + 8 a2

]

z+−2 :=

[

0i a2

−128 + 8 a2

]

4.3.3 The Transformation into Action Angle Coordinates

As mentioned above, the aim of this section 4.3 is to derive explicit formulae for the maps g2k,n whichappear in the equation for the action variable h of the restricted system (cf. (1.158)). Considering thedefinition of these maps g2k,n via (1.156), i.e.

g2k,n(h) =∑

k1,k2∈Z

k1+k2=k

∑

|n1|≤N,|n2|≤2Nn1+n2=n

(

F1,1k1,n1,3

(h)S1k2,n2

(h))

+ F2,0k,n,3(h) (4.36)

we see that we need to compute the maps F1,1k1,n1,3

, S1k2,n2

(h) and F2,0k1,n1,3

. In this subsection we aim

on explicit formulae for F1,1k1,n1,3

and F2,0k1,n1,3

. The computation of the quantities S1k,n is related to the

calculation of the attractive invariant manifold, which we postpone until the next subsection. Howeverwe will prepare these computations as well, giving some formulae for G1,0

k,n. Note that by consequence of

remark 1.6.11 we have g1k,n(h) = 0 for all h ∈ R as the map F 1 considered here vanishes for η = 0.

Recall that by (1.98)

Φ(ϕ, h) := (q, p)(ϕ,P(h)) := (q, p)( ϕΩ(P(h)) ; 0,P(h)) (4.37)


where (q, p)(t; 0, p0) are the solutions of (4.16) given by (4.17). Using (4.34) and (4.35) derived abovedefinition 1.6.5 yields

F1,13 (t, ϕ, h)H =

1ddhH(0,P(h))

(

∇H(Φ(ϕ, h))∣

∣

∣

[

ζ+1

(

eiq(ϕ,P(h)) − 1)

+ ζ−1

(

e−iq(ϕ,P(h)) − 1)

]

H

)

eit

+ 1ddhH(0,P(h))

(

∇H(Φ(ϕ, h))∣

∣

∣

[

ζ+−1

(

eiq(ϕ,P(h)) − 1)

+ ζ−−1

(

e−iq(ϕ,P(h)) − 1)

]

H

)

e−it (4.38)

F2,03 (t, ϕ, h) = − 1

ddhH(0,P(h))

(

∇H(Φ(ϕ, h))

∣

∣

∣

∣

(

0p(ϕ,P(h))

))

− (m+ ) 1ddhH(0,P(h))

(

∇H(Φ(ϕ, h))

∣

∣

∣

∣

(

0

1−(

eiq(ϕ,P(h)) + e−iq(ϕ,P(h)))

/2

))

+ 1ddhH(0,P(h))

(

∇H(Φ(ϕ, h))∣

∣

∣

[

z00 + z+0 eiq(ϕ,P(h)) + z−0 e

−iq(ϕ,P(h))]

)

+ 1ddhH(0,P(h))

(

∇H(Φ(ϕ, h))∣

∣

∣

[

z02 + z+2 eiq(ϕ,P(h)) + z−2 e

−iq(ϕ,P(h))]

)

ei2t

+ 1ddhH(0,P(h))

(

∇H(Φ(ϕ, h))∣

∣

∣

[

z0−2 + z+−2 eiq(ϕ,P(h)) + z−−2 e

−iq(ϕ,P(h))]

)

e−i2t.

(4.39)

The analogous result may be obtained for G1,0k,n from (4.26), (1.138) and (1.143)

G1,0(t, ϕ, h) =(

eiq(ϕ,P(h)) − 1)

v eit +(

e−iq(ϕ,P(h)) − 1)

v e−it. (4.40)

In a next step we introduce some Fourier series which will enable us to express F1,03 , F2,0

3 and G1,0 inaccordance with (1.142), (1.143).

Let (ak)k∈Z, (bk)k∈Z

, (αk)k∈Z, (βk)k∈Z

, (αβk)k∈Z, (aβk)k∈Z

and (bβk)k∈Zbe the unique sequences of

maps defined on R, such that

eiq(ϕ,P(h)) =∑

k∈Z

ak(h) eikϕ p(ϕ,P(h)) =

∑

k∈Z

bk(h) eikϕ

(

eiq(ϕ,P(h)) − 1)

=∑

k∈Z

αk(h) eikϕ ∇H(Φ(ϕ, h)) =

∑

k∈Z

βk(h) eikϕ (4.41)

(

eiq(ϕ,P(h)) − 1)

∇H(Φ(ϕ, h)) =∑

k∈Z

αβk(h) eikϕ eiq(ϕ,P(h)) ∇H(Φ(ϕ, h)) =

∑

k∈Z

aβk(h) eikϕ

p(ϕ,P(h))∇H(Φ(ϕ, h)) =∑

k∈Z

bβk(h) eikϕ

hold for all h, ϕ ∈ R. Note that ak(h), bk(h), αk(h) ∈ C, while βk(h), αβk(h), aβk(h), bβk(h) ∈ C 2.Moreover we use double letters to denote the quantities αβ, aβ which may be unconventional for thereader. However, we will see in what follows that these sequences may be represented as the convolution oftwo sequences (in fact the sequences (αk)k∈Z

, (βk)k∈Zand (ak)k∈Z

, (βk)k∈Z, respectively). This justifies

the special notation.


Taking the complex conjugate of (4.3.3) it is evident that

(

e−iq(ϕ,P(h)) − 1)

=∑

k∈Z

αk(h)e−ikϕ

(

e−iq(ϕ,P(h)) − 1)

∇H(Φ(ϕ, h)) =∑

k∈Z

αβk(h)e−ikϕ

e−iq(ϕ,P(h)) ∇H(Φ(ϕ, h)) =∑

k∈Z

aβk(h)e−ikϕ.

With the help of these representations we now are in the position to rewrite (4.38)–(4.40) as follows:

F1,13 (t, ϕ, h)H := 1

ddhH(0,P(h))

[(

∑

k∈Z

αβk(h) eikϕ

∣

∣

∣

∣

∣

ζ+1 H

)

+

(

∑

k∈Z

αβk(h)e−ikϕ

∣

∣

∣

∣

∣

ζ−1 H

)]

eit

+ 1ddhH(0,P(h))

[(

∑

k∈Z

αβk(h) eikϕ

∣

∣

∣

∣

∣

ζ+−1 H

)

+

(

∑

k∈Z

αβk(h)e−ikϕ

∣

∣

∣

∣

∣

ζ−−1 H

)]

e−it

hence in correspondance with (1.142)

F1,1k,1,3(h)H = 1

ddhH(0,P(h))

[

(

αβk(h)| ζ+1 H)

+(

αβ−k(h)∣

∣

∣ζ−1 H

)

]

F1,1−k,−1,3(h)H = 1

ddhH(0,P(h))

[

(

αβ−k(h)∣

∣ ζ+−1 H)

+(

αβk(h)∣

∣

∣ ζ−−1 H

)

]

F1,0k1,n1,3

= 0 else.

(4.42)

In much the same way we find

F2,03 (t, ϕ, h) := − 1

ddhH(0,P(h))

(

∑

k∈Z

bβk(h) eikϕ

∣

∣

∣

∣

∣

(

01

)

)

−(m+ ) 1ddhH(0,P(h))

[(

∑

k∈Z

βk(h) eikϕ

∣

∣

∣

∣

∣

(

01

)

)

−(

∑

k∈Z

aβk(h) eikϕ

∣

∣

∣

∣

∣

(

012

)

)

−(

∑

k∈Z

aβk(h)e−ikϕ

∣

∣

∣

∣

∣

(

012

)

)]

+ 1ddhH(0,P(h))

[(

∑

k∈Z

βk(h) eikϕ

∣

∣

∣

∣

∣

z00

)

+

(

∑

k∈Z

aβk(h) eikϕ

∣

∣

∣

∣

∣

z+0

)

+

(

∑

k∈Z

aβk(h)e−ikϕ

∣

∣

∣

∣

∣

z−0

)]

+ 1ddhH(0,P(h))

[(

∑

k∈Z

βk(h) eikϕ

∣

∣

∣

∣

∣

z02

)

+

(

∑

k∈Z

aβk(h) eikϕ

∣

∣

∣

∣

∣

z+2

)

+

(

∑

k∈Z

aβk(h)e−ikϕ

∣

∣

∣

∣

∣

z−2

)]

ei2t

+ 1ddhH(0,P(h))

[(

∑

k∈Z

βk(h) eikϕ

∣

∣

∣

∣

∣

z0−2

)

+

(

∑

k∈Z

aβk(h) eikϕ

∣

∣

∣

∣

∣

z+−2

)

+

(

∑

k∈Z

aβk(h)e−ikϕ

∣

∣

∣

∣

∣

z−−2

)]

e−i2t


hence

F2,0k,2,3(h) =

1ddhH(0,P(h))

[

(

βk(h)| z02)

+(

aβk(h)| z+2)

+(

aβ−k(h)∣

∣

∣ z−2

)

]

F2,0−k,−2,3(h) =

1ddhH(0,P(h))

[

(

β−k(h)| z0−2

)

+(

aβ−k(h)∣

∣ z+−2

)

+(

aβk(h)∣

∣

∣ z−−2

)

]

F2,0k,0,3(h) =

1ddhH(0,P(h))

[

(

βk(h)| z00)

+(

aβk(h)| z+0)

+(

aβ−k(h)∣

∣

∣ z−0

)

−

(

bβk(h)

∣

∣

∣

∣

(

01

))

− (m+ )

((

βk(h)

∣

∣

∣

∣

(

01

))

−(

aβk(h) + aβ−k(h)

∣

∣

∣

∣

(

012

)))

]

F2,0k,n(h) = 0 else.

(4.43)

Finally (4.40) and (4.3.3) imply

G1,0(t, ϕ, h) =∑

k∈Z

αk(h) v ei(kϕ+t) +

∑

k∈Z

αk(h) v e−i(kϕ+t). (4.44)


4.3.4 The Attractive Invariant Manifold

Using the results found in proposition 1.6.7 together with (4.44) we can immediately write down theexplicit formula for the coefficient maps Sjk,n :

S1k,1(h) = [i(k ω(h) + 1)IR3 −A]

−1αk(h) v

S1−k,−1(h) = [i(−k ω(h)− 1)IR3 −A]

−1αk(h) v

S1k,n(h) = 0 if |n| 6= 1

(4.45)

where ω(h) = Ω(P(h)) (cf. (1.99)). As mentioned in the introduction of section 4.3.3 we may confineourselves to the explicit computation of these coefficients.

In what follows we will list the Maple [15] –procedures corresponding to the quantities discussed inorder to enable the reader to reproduce the results found here. We start with the procedure used todefine S1

k,1, S1−k,−1 (for |k| smaller than a given integer M):

S1_build := proc(kappa,alpha,alphacc)

local k,SS;global Omega,M,v,vb,a;

SS[1] := table([seq(k = scalarmul(multiply(inverse(scalarmul(Id3,I*(k*Omega(kappa,a)+1))-A),v),alpha[k]),k = -M .. M)]);

SS[-1] := table([seq(-k = scalarmul(multiply(inverse(scalarmul(Id3,I*(-k*Omega(kappa,a)-1))-A),vb),alphacc[k]),k = -M .. M)]);

table([seq((k,-1) = evalm(SS[-1][k]),k = -M .. M),seq((k,1) = evalm(SS[1][k]),k = -M .. M)])

end

(The variable ”kappa” will be defined in definition 4.3.2. The expression ”alphacc” symbolizes theconjugate complex of ”alpha”.)


4.3.5 Main Result of Section 4.3

Combining the results derived in the preceding two subsections we now are in the position to evaluatethe formula given in (4.38) for g2k,n. We will see that due to the theory deduced in chapter 2 we mayrestrict ourselves to the cases where (k, n) = (0, 0) or n < 0 < k.

In a first step we collect the terms in (4.43)–(4.45) which contribute to the sum appearing in (4.36) for(k, n) = (0, 0):

g20,0(h) =∑

k∈Z

(

F1,1k,n,3(h)S1

−k,−1(h) + F1,1−k,−n,3(h)S1

k,1(h))

+ F2,00,0,3(h)

= 1ddhH(0,P(h))

(

∑

k∈Z

[

(

αβk(h)| ζ+1 S1−k,−1(h)

)

+(

αβ−k(h)∣

∣

∣ζ−1 S1

−k,−1(h))

+(

αβ−k(h)∣

∣ ζ+−1 S1k,1(h)

)

+(

αβk(h)∣

∣

∣ζ−−1 S1

k,1(h))

]

+(

β0(h)| z00)

+(

aβ0(h)| z+0)

+(

αβ0(h)∣

∣

∣ z−0

)

−

(

bβ0(h)

∣

∣

∣

∣

(

01

))

− (m+ )

((

β0(h)

∣

∣

∣

∣

(

01

))

−(

aβ0(h) + aβ0(h)

∣

∣

∣

∣

(

012

)))

)

. (4.46)

evalG00 := proc(beta, alphabeta, abeta, bbeta, alphabetacc, abetacc, bbetacc, zeta, z, S)

local G00, k;global apar, M, cutoff, rho, m;

G00 := 0;for k from -M to M do

G00 := G00+ evalf(dotprod(alphabeta[k], multiply(zeta[1][1], S[-k, -1]),’orthogonal’))+ evalf(dotprod(alphabetacc[-k],multiply(zeta[-1][1], S[-k, -1]), ’orthogonal’))+ evalf(dotprod(alphabeta[-k], multiply(zeta[1][-1], S[k, 1]),’orthogonal’))+ evalf(dotprod(alphabetacc[k],multiply(zeta[-1][-1], S[k, 1]), ’orthogonal’))

od;G00 := G00 + normal(dotprod(beta[0], z[0][0], ’orthogonal’), expanded)

+ normal(dotprod(abeta[0], z[1][0], ’orthogonal’),expanded)+ normal(dotprod(abetacc[0], z[-1][0], ’orthogonal’), expanded);

G00 := G00 - rho*normal(dotprod(bbeta[0], vector([0, 1]), ’orthogonal’), expanded);G00 := G00 - (m + rho)*normal(dotprod( evalm(beta[0] - 1/2*abeta[0] - 1/2*abetacc[0]),

vector([0, .5]), ’orthogonal’), expanded);if abs(Im(G00)) < cutoff then G00 := evalc(Re(G00))else print(‘## E R R O R : ## cutoff too small in evalG00(..) : Im(G00)=‘, Im(G00))fi;G00

end


Carrying out the same step in the case when k 6= 0, n < 0, one gets the following expression:

g2−k,−2(h) =∑

k1,k2∈Z

k1+k2=k

(

F1,1−k1,−1,3(h)S1

−k2,−1(h))

+ F2,0−k,−2,3(h)

= 1ddhH(0,P(h))

(

∑

k1,k2∈Z

k1+k2=k

[

(

αβ−k1(h)∣

∣ ζ+−1 S1−k2,−1(h)

)

+(

αβk1(h)∣

∣

∣ ζ−−1 S1−k2,−1(h)

)

]

+(

β−k(h)| z0−2

)

+(

aβ−k(h)∣

∣ z+−2

)

+(

αβk(h)∣

∣

∣ z−−2

)

)

(4.47)

and g2k,n = 0 if n < 0 and n 6= −2.

evalGk2 := proc(k, beta, alphabeta, abeta, alphabetacc, abetacc, zeta, z, S)

local Gk2, k1, k2;global M, cutoff;

Gk2 := 0;for k1 from -M to M dofor k2 from -M to M do

if k1 + k2 = k thenGk2 := Gk2+ evalf(dotprod(alphabeta[-k1], evalm(multiply(zeta[1][-1],S[-k2, -1])),

’orthogonal’));+ evalf(dotprod(alphabetacc[k1], evalm(multiply(zeta[-1][-1],S[-k2, -1])),

’orthogonal’))fi

od;od;Gk2 := Gk2 + normal(dotprod(beta[-k], z[0][-2], ’orthogonal’), expanded);Gk2 := Gk2 + normal(dotprod(abeta[-k], z[1][-2], ’orthogonal’), expanded);Gk2 := Gk2 + normal(dotprod(abetacc[k], z[-1][-2], ’orthogonal’), expanded);2*abs(Gk2)

end


4.3.6 The Calculation of the Fourier Coefficients

We have seen in the preceeding chapters that the maps g20,0 and g2km,−2 play a crucial role in the analysisof the system under investigation. In order to apply the results derived there we need more informationabout the properties of these maps g20,0 and g2−k,−2. Due to the complexity of the explicit formulae (4.46)and (4.47) we will gain this information by approximating these expressions numerically. Aiming on suchnumerical evaluations it is necessary to find a way to compute the values βk(h), αβk(h), aβk(h) andbβk(h) arising in (4.46) and (4.47). This is the purpose of this last subsection. As we will deal withvarious sequences of Fourier–coefficients in what follows, we introduce the following notation.

Definition 4.3.1 The convolution x = (xk)k∈Zof the sequences y = (yk)k∈Z

, z = (zk)k∈Z(where yk ∈ C

and zk ∈ Cn, n ∈ N∗) is defined as follows:

x = y ∗ z :=

∑

k1,k2∈Z

k1+k2=k

yk1zk2

k∈Z

. (4.48)

With the help of convolutions it will be possible to express the maps βk(h), αβk(h), aβk(h) and bβk(h)in an easy way. Let us therefore introduce the following sequences:

Definition 4.3.2 In accordance with the notation found in [2] we introduce the following abbreviations

κ(h) :=P(h)

aq(h) := e−π

K(√

1−κ(h)2)K(κ(h)) (4.49)

and define then sequences dn(h) = (dn(h)k)k∈Z, iκsn(h) = (iκsn(h)k)k∈Z

and cn(h) = (cn(h)k)k∈Zby

dn(h)k =

π

K(κ(h))

q(h)|k|/2

1 + q(h)|k|if k even

0 else

iκsn(h)k =

sgn(k)π

K(κ(h))

q(h)|k|/2

1− q(h)|k|if k odd

0 else

cn(h)k =

π

κ(h)K(κ(h))

q(h)|k|/2

1 + q(h)|k|if k odd

0 else

.

(4.50)

Note that considering the central domain, i.e. J = JC the limit h→ ∞ corresponds to P(h) → a henceκ → 1. If we consider regions in JC close to the border LJr we therefore focus on values κ / 1. Onthe other hand this border LJr is close to the separatrices of the unperturbed pendulum (cf. figure 1.2).Hence the values κ / 1 correspond to regions close to the separatrices. This is true for the cases J = JLand J = JU as well.


K := LegendreKc q := (κ, pot ) → e

(

−

pot π LegendreKc1( κ )LegendreKc(κ )

)

Ω := (κ, a ) → evalf

(

1

4

π a

K(κ )

)

dn_build := proc(kappa)

local k,dn;global K,q,M;

dn := table([seq(k = 0,k = -M .. M)]);for k from -M to M do

if type(k,even) thendn[k] := evalf(Pi/K(kappa)*q(kappa,1/2*abs(k))/(1+q(kappa,abs(k))))

fiod;table([seq(k = dn[k],k = -M .. M)])

end

iksn_build := proc(kappa)

local k,iksn;global K,q,M;

iksn := table([seq(k = 0,k = -M .. M)]);for k from -M to M do

if type(k,odd) theniksn[k] := evalf(signum(k)*Pi/K(kappa)*q(kappa,1/2*abs(k))/(1-q(kappa,abs(k))))

fiod;table([seq(k = iksn[k],k = -M .. M)])

end

cn_build := proc(kappa)

local k,cn;global K,q,M;

cn := table([seq(k = 0,k = -M .. M)]);for k from -M to M do

if type(k,odd) thencn[k] := evalf(Pi/kappa/K(kappa)*q(kappa,1/2*abs(k))/(1+q(kappa,abs(k))))

fiod;table([seq(k = cn[k],k = -M .. M)])

end


In a first step we express the quantities ak(h) :

Lemma 4.3.3 Recall the Fourierseries eiq(ϕ,P(h)) =∑

k∈Z

ak(h) eikϕ introduced in (4.3.3). The series

a(h) := (ak(h))k∈Zis given by

a(h) = (dn(h) + iκsn(h)) ∗ (dn(h) + iκsn(h)) . (4.51)

PROOF: As explained in (4.17) the solution (q(t; 0, p0), p(t; 0, p0)) of system (4.16) which is used in(4.37) to define the maps q(ϕ,P(h)), p(ϕ,P(h)) may be expressed using the Jacobian Elliptic Function :

sin(12q(t; 0, p0)) =p0a

sn(a

2t;p0a

)

p(t; 0, p0) = p0 cn(a

2t;p0a

)

.

On one hand the second equation of (4.16) implies therefore

−(a

2

)2

sin(q(t; 0, p0)) =d

dtp(t; 0, p0) =

d

dt

(

p0 cn(a

2t;p0a

))

BF=

731.02−a2p0 sn

(a

2t;p0a

)

dn(a

2t;p0a

)

.

(Here and in all subsequent sections the notationBF=

731.02refers to the corresponding formula given in [2]).

Taking this last equation and applying (4.37) yields

sin(q(ϕ,P(h))) = sin(q( ϕΩ(P(h)) ; 0,P(h)))

= 2a P(h) sn

(

a2

ϕΩ(P(h)) ;

P(h)a

)

dn(

a2

ϕΩ(P(h)) ;

P(h)a

)

= 2 κ(h) sn(

a2

ϕΩ(P(h)) ;

P(h)a

)

dn(

a2

ϕΩ(P(h)) ;

P(h)a

)

.

Thus by (4.18)

sin(q(ϕ,P(h))) = 2 κ(h) sn(

ϕ2π4K(κ(h));κ(h)

)

dn(


)

. (4.52)

On the other hand we use the identity

P(h)2

2= H(0,P(h)) = H(q(t; 0,P(h)), p(t; 0,P(h))) =

p(t; 0,P(h))2

2+(a

2

)2

(1 − cos(q(t; 0,P(h))))

to express cos(q(t; 0,P(h))) as follows:

cos(q(t; 0,P(h))) = 1 +

(

2

a

)2(

p(t; 0,P(h))2

2− P(h)

2

2

)

= 1 +

(

2

a

)2 P(h)2

2

(

cn(a

2t;κ(h)

)2

− 1

)

= 1 + 2 κ(h)2

(

cn(a

2t;κ(h)

)2

− 1

)

BF=

121.001− 2 κ(h)2 sn

(a

2t;κ(h)

)2

,


which again by (4.18) implies

cos(q(ϕ,P(h))) = 1− 2 κ(h)2sn(


)2. (4.53)

Using Eulers equation we write (4.52) and (4.53) in complex form :

eiq(ϕ,P(h)) = cos(q(ϕ,P(h))) + i sin(q(ϕ,P(h)))

= 1− 2 κ(h)2 sn(


)2

+ i 2 κ(h) sn(


)

dn(


)

BF=

121.00κ(h)

2sn(


)2+ dn

(


)2

− 2 κ(h)2sn(


)2

+ i 2 κ(h) sn(


)

dn(


)

=(

dn(


)

+ i κ(h) sn(


)

)2

such that

eiq(ϕ,P(h))/2 = dn(


)

+ i κ(h) sn(


)

. (4.54)

Since the Fourier Series of the Jacobian Elliptic Functions are known (cf. [2] formulae 908) we are ableto find the corresponding series for eiq(ϕ,P(h))/2. According to the tables given in [2] let therefore q(h) beas defined in (4.49). Then

dn(


) BF=

908.03

π2K(κ(h)) +

2πK(κ(h))

∑

m≥0

q(h)m+1

1+q(h)2(m+1) cos(

(m+ 1) πK(κ(h))

ϕ2π4K(κ(h))

)

= π2K(κ(h)) +

2πK(κ(h))

∑

m≥0

q(h)m+1

1+q(h)2(m+1) cos (2(m+ 1)ϕ)

= π2K(κ(h)) +

πK(κ(h))

∑

m≥0

q(h)m+1

1+q(h)2(m+1)

(

ei2(m+1)ϕ + e−i2(m+1)ϕ)

= π2K(κ(h)) +

πK(κ(h))

∑

k>0even

q(h)k/2

1+q(h)k

(

eikϕ + e−ikϕ)

and in a similar way

i κ(h) sn(


) BF=

908.01iκ(h) 2π

κ(h)K(κ(h))

∑

m≥0

q(h)m+1/2

1−q(h)2m+1 sin(

(2m+ 1) π2K(κ(h))

ϕ2π4K(κ(h))

)

= i 2πK(κ(h))

∑

m≥0

q(h)m+1/2

1−q(h)2m+1 sin ((2m+ 1)ϕ)

= πK(κ(h))

∑

m≥0

q(h)m+1/2

1−q(h)2m+1

(

ei(2m+1)ϕ − e−i(2m+1)ϕ)

= πK(κ(h))

∑

k>0odd

q(h)k/2

1−q(h)k(

eikϕ − e−ikϕ)

.

These identities correspond to definition 4.3.2 of the coefficients dn(h)k and iκsn(h)k. It then followsfrom (4.54) that

eiq(ϕ,P(h))/2 =∑

k∈Z

(dn(h) + iκsn(h))k eikϕ =

∑

k∈Z

(dn(h)k + iκsn(h)k) eikϕ.


For the expansion (4.42) we thus find

∑

k∈Z

ak(h) eikϕ = eiq(ϕ,P(h)) = eiq(ϕ,P(h))/2 · eiq(ϕ,P(h))/2

=

(

∑

k∈Z

(dn(h) + iκsn(h))k eikϕ

)

·(

∑

k∈Z

(dn(h) + iκsn(h))k eikϕ

)

=∑

k1,k2∈Z

(dn(h) + iκsn(h))k1 (dn(h) + iκsn(h))k2 ei(k1+k2)ϕ

such that

ak(h) =∑

k1,k2∈Z

k1+k2=k

(dn(h) + iκsn(h))k1(dn(h) + iκsn(h))k2

= ((dn(h) + iκsn(h)) ∗ (dn(h) + iκsn(h)))k

i.e. a = (dn(h) + iκsn(h)) ∗ (dn(h) + iκsn(h)) as claimed. This completes the proof of lemma 4.3.3.

a_build := proc(dn,iksn)

local u,k;global M;

u :=table([seq(k = dn[k]+iksn[k],k = -M .. M)]);ComplexFold(u,u)

end

The next lemma provides similar formulae for the Fourier series p(ϕ,P(h)) =∑

k∈Z

bk(h) eikϕ.

Lemma 4.3.4 The series b(h) := (b(h)k)k∈Zof Fourier coefficients for p(ϕ,P(h)) =

∑

k∈Z

bk(h) eikϕ is

given by

b(h) = P(h) cn(h) = a κ(h) cn(h), (4.55)

where cn(h) denotes the sequence defined in (4.51).

PROOF: From the definitions (4.37), (4.3.3) together with (4.17) we have

∑

k∈Z

bk(h) eikϕ = p(ϕ,P(h)) = p( ϕ

Ω(P(h)) , 0,P(h)) = P(h) cn(

a2

ϕΩ(P(h)) ;

P(h)a

)

= P(h) cn(


)

.


Rewriting the Fourier series 908.02 in [2] yields

cn(


)

= 2πκ(h)K(κ(h))

∑

m≥0

q(h)m+1/2

1+q(h)2m+1 cos(

(2m+ 1) π2K(κ(h))

ϕ2π4K(κ(h))

)

= 2πκ(h)K(κ(h))

∑

m≥0

q(h)m+1/2

1+q(h)2m+1 cos ((2m+ 1)ϕ)

=π

κ(h)K(κ(h))

∑

m≥0

q(h)m+1/2

1+q(h)2m+1

(

ei(2m+1)ϕ + e−i(2m+1)ϕ)

=π

κ(h)K(κ(h))

∑

k>0odd

q(h)k/2

1+q(h)k

(

eikϕ + e−ikϕ)

thus for cn(h)k as defined in (4.50), we obtain bk(h) = P(h) cn(h)k as claimed.

b_build := proc(kappa,cn)

local k;global a,M;

table([seq(k = evalf(a*kappa*cn[k]),k = -M .. M)])

end

With the help of the maps ak(h), bk(h) it now is possible to express βk(h), αβk(h), aβk(h) and bβk(h),using appropriate convolutions again.

Lemma 4.3.5 The sequences of coefficient maps defined in (4.3.3) are given by the following identities :

α(h) = a(h)− (. . . , 0, 1, 0, . . . ) β(h) =

(

−i a22 (iκsn(h) ∗ dn(h))b(h)

)

αβ(h) = α(h) ∗ β(h) aβ(h) = a(h) ∗ β(h) bβ(h) = b(h) ∗ β(h). (4.56)

PROOF: Recalling∑

k∈Z

αk(h) eikϕ = eiq(ϕ,P(h)) − 1 =

∑

k∈Z

ak(h) eikϕ − 1 =

∑

k 6=0

ak(h) eikϕ + (a0(h)− 1) ,

the first identity follows at once. Using equation (4.52) we write

(a

2

)2

sin(q(ϕ,P(h))) = −2 i(

a2

)2 ( i2 sin(q(ϕ,P(h)))

)

= −2 i(

a2

)2 (i κ(h) sn

(


)

dn(


))

= −2 i(

a2

)2

(

∑

k∈Z

iκsnk(h) eikϕ

) (

∑

k∈Z

dnk(h) eikϕ

)

= −2 i(

a2

)2 ∑

k∈Z

(iκsn(h) ∗ dn(h))k eikϕ,


implying

∑

k∈Z

βk(h) eikϕ = ∇H(Φ(ϕ, h)) =

((

a2

)2sin(q(ϕ,P(h)))p(ϕ,P(h))

)

=

−2 i(

a2

)2 ∑

k∈Z

(iκsn(h) ∗ dn(h))k eikϕ∑

k∈Z

bk(h) eikϕ

.

Thus the second statement follows immediately by comparing coefficients of eikϕ, k ∈ Z. The proof ofthe remaining identities is a simple consequence of (4.3.3) and therefore omitted.

alpha_build := proc(a)

local k,alpha;global M;

alpha := table([seq(k = a[k],k = -M .. M)]);alpha[0] := evalf(a[0]-1);table([seq(k = alpha[k],k = -M .. M)])

end

beta_build := proc(iksn,dn,b)

local iksndn,k;global M,apar;

iksndn := ComplexFold(iksn,dn);table([seq(k = vector([evalf(-1/2*I*apar^2*iksndn[k]),b[k]]),k = -M .. M)])

end

alphabeta_build := proc(alpha,beta)

local beta1,beta2,k,bg1,bg2;global M;

beta1 :=table([seq(k = beta[k][1],k = -M .. M)]);beta2 :=table([seq(k = beta[k][2],k = -M .. M)]);bg1 := ComplexFold(alpha,beta1);bg2 := ComplexFold(alpha,beta2);table([seq(k = vector([bg1[k],bg2[k]]),k = -M .. M)])

end

abeta_build := proc(a,beta)

local beta1,beta2,k,bg1,bg2;global M;

beta1 :=table([seq(k = beta[k][1],k = -M .. M)]);beta2 :=table([seq(k = beta[k][2],k = -M .. M)]);bg1 := ComplexFold(a,beta1);bg2 := ComplexFold(a,beta2);table([seq(k = vector([bg1[k],bg2[k]]),k = -M .. M)])

end


bbeta_build := proc(b, beta)

local beta1, beta2, k, bg1, bg2;global M;

beta1 := table([seq(k = beta[k][1], k = -M .. M)]);beta2 := table([seq(k = beta[k][2], k = -M .. M)]);bg1 := ComplexFold(b, beta1);bg2 := ComplexFold(b, beta2);table([seq(k = vector([bg1[k], bg2[k]]), k = -M .. M)])

end

alphacc_build := proc(alpha)

local k;global M;

table([seq(k = conjugate(alpha[k]),k = -M .. M)])

end

alphabetacc_build := proc(alphabeta)

local k;global M;

table([ seq(k = vector([conjugate(alphabeta[k][1]), conjugate(alphabeta[k][2])]),k = -M .. M)])

end

abetacc_build := proc(abeta)

local k;global M;

table([seq(k = vector([conjugate(abeta[k][1]), conjugate(abeta[k][2])]), k = -M .. M)])

end

bbetacc_build := proc(bbeta)

local k;global M;

table([seq( k = vector([conjugate(bbeta[k][1]), conjugate(bbeta[k][2])]), k = -M .. M)])

end

4.4. Preliminary Remarks on Numerical Evaluation 163

4.4 Preliminary Remarks on Numerical Evaluation

4.4.1 The Choice of the Parameters ε and a

Let us recall that the maps g20,0 and g2km,−2 do not depend on the perturbation parameter ε but on thesingle parameter a.

We therefore will not find ourselves in the position to discuss the choice for ε under which the numericaland theoretical results match. The results found here are independent on ε and valid provided that ε ischosen sufficiently small for the theoretical considerations carried out in the first three chapters.

By way of contrast the computations carried out numerically depend strongly on the value chosen for theparameter a. We therefore will present results for a variety of choices of this parameter. As the parametera varies in [0.54, 20.38] in technical considerations we will choose the values a = 0.54 and a = 20.38 whenperforming the actual numerical evaluation. As we have found an interesting behaviour of the solutionsfor a = 4.1 in the numerical simulations described in section 4.2.2 we will examine this value for a aswell.

4.4.2 The Independence on the Map P

The qualitative behaviour of system (1.158) does not depend on the choice of the map P considered in1.97 a–1.97d. (The main reason for introducing P was to handle regularity problems hence a technicalmatter (cf. section 1.3.4).) For the numerical results to be independent of P we proceed as follows:

Recalling the formulae (4.45)–(4.47) for g20,0, g2km,−2 and S1

±k,±1 we see that the quantities

ddhH(0,P(h)) g20,0(h) and d

dhH(0,P(h)) g2km,−2(h)

may be expressed in terms of ω(h), constant matrices and vectors as well as the Fourier coefficient mapsαk(h), βk(h), αβk(h), aβk(h) and bβk(h). Moreover it follows from the formulae found in section 4.3.6and particularly (4.49), (4.50) that these coefficient maps depend on h via the function q(h), hence via

κ(h) = P(h)a . Therefore there exist maps G0,0 and G−k,−2, independent on P and defined for κ ∈ [0, 1),

satisfying

G0,0(κ(h)) =ddhH(0,P(h)) g20,0(h) and G−k,−2(κ(h)) = 2

∣

∣

ddhH(0,P(h)) g2km,−2(h)

∣

∣ . (4.57)

We therefore will compute the values of G0,0, G−k,−2, avoiding to fix a map P . Note that in a res-onance h = hm it follows from the last identities together with (2.30) that the inequalities |a0| >√

(ac1)2+ (as1)

2(cf. lemma 2.3.4) and |a0| <

√

(ac1)2+ (as1)

2(cf. lemma 2.3.5) respectively are equiv-

alent to |G0,0(κ(hm))| > G−k,−2(κ(hm)) and |G0,0(κ(hm))| < G−k,−2(κ(hm)). Thus the comparison of|G0,0(κ(hm))| with G−k,−2(κ(hm)) enables us to decide if either of the situations discussed in section 2.3.3and section 2.3.4 applies, i.e. if all solutions or only most of the solutions pass the resonance.

For the discussion in the outer zones (cf. proposition 2.3.1) it suffices to gain information on the sign ofg20,0(h). From the identities (4.49), (4.57) together with the explicit form (4.1.3) of the Hamiltonian thisis given by

g20,0(h) =G0,0(κ(h))ddhP(h)a κ(h)

.


Since ddhP(h) > 0 for all h 6= 0 we therefore obtain this information on the sign of g20,0(h) by considering

the plot of the map κ 7→ G0,0(κ)aκ (and κ 7→ ±G0,0(κ)

a/κ in the upper, lower domain respectively). Moreover,

in the central domain, this map provides information on the stability of h = 0 in the following way:

From lemma 1.6.9 and corollary 3.2.3 (see also section 4.6) it follows that

g20,0(h) = P(h)ddhP(h)

g2,10,0 +O( P(h)2

ddhP(h)

)

thus

g2,10,0 = limh→0

ddhP(h) g20,0(h)

P(h)= limκ→0

G0,0(κ)

(κ a)2=

1

a

d

dκ

(

G0,0(κ)

a κ

)

(0).

The valuesG0,0(κ)

(κa)2for small κ therefore approximate the slope of the map

G0,0(κ)aκ at κ = 0 and correspond

to the quantity g2,10,0 necessary to discuss the stability of h = 0. The evaluation ofG0,0(κ)

(κa)2will be printed

as well.


4.4.3 How to Determine Resonances

Let us first note that by consequence of (4.36), (4.38), (4.43) and (4.45) the maps g2k,n vanish if |n| 6∈ 0, 2.Thus the set R of resonant frequencies as defined in GA 2.2 is given by

R =

q ∈ Q | q ∈ [Ω(Jr),Ω(0)], ∃k ∈ Z : q = −2k

=

q ∈ Q

∣

∣

∣

∣

∣

q ∈[

a

2

2π

4K(Jra ),a

2

]

, ∃k ∈ N∗ : q = 2k

Hence by solving the equations Ω(κ, a) = a2

2π4K(κ) =

2k with respect to κ we obtain a family

(km, κkm) |m = 1..M of solutions such that the resonances h ∈ H appearing here are given by κ(hm) =κkm .

As the Fourier coefficients g2k,n are of size O(1/k3) (cf. remark 1.6.10) we determine the resonances for

|k| ≤ kmax only. The value kmax ∈ N∗ is chosen in a way such that the corresponding values 2∣

∣

∣g2k,n(κkm)

∣

∣

∣

are smaller than g20,0(κkm), hence passage through resonance takes place for all the resonances with indicesk ≥ kmax (cf. section 2.3.4).

DetectResonances := proc(kmax,kappa1,kappa2)

local Resonances,Res,k,kappa,j;global Omega,apar;

j := 0;Resonances := array(1 .. kmax);printf(‘******************* Detection of Resonances ***********************‘);lprint();for k to kmax do

Res := fsolve(Omega(kappa,apar) = 2/k,kappa,kappa1 .. kappa2);if type(Res,float) then

j := j+1;Resonances[j] := [k,Res];printf(‘2 : %g-Resonance in %g......‘,k,Res);lprint()

fiod;printf(‘******************* DETECTION COMPLETE *\**********************‘);lprint();[j,table([seq(k = Resonances[k],k = 1 .. j)])]

end

In figure 4.17 a plot of the map Ω over κ and a is shown. The level curves Ω = 2k found via the procedure

”DetectResonances” are depicted in figure 4.18 : The curves for Ω = 2 : 1, Ω = 2 : 2 and Ω = 2 : 3 (mostright to left) are clearly visible, while the curves for larger k approache the level curve Ω = 0 (i.e. a = 0).


0

1

2

3

4

5

abar

0

0.2

0.4

0.6

0.8

1

kappa

0

0.5

1

1.5

2

2.5

omega(abar, kappa)

Figure 4.17: 3D–Graph of Ω(κ, a)

0

0.2

0.4

0.6

0.8

1

kappa

0 1 2 3 4 5abar

Figure 4.18: Results of procedure ”DetectResonances”

Remark 4.4.1 The results of the procedure ”DetectResonances” correspond to the level curves of thethree dimensional plot of Ω(κ, a). Taking into account that Ω(0, a) = a

2 is the maximal value of Ω we seethat there exist basically three cases:

4.58 a. 0 < a < 1 The values of Ω are bounded by 2 : 4. Hence all the resonances appearing are higher orderresonances 2 : 5, 2 : 6, . . . . In view of the bounds given in lemma 1.6.10 we expect assumption(2.33) of remark 2.3.4 to be satisfied. Hence for these values of a passage through resonance isprobable.

4.58 b. 1 < a ≤≈ 5 This is the most difficult range for the qualitative discussion. The resonances κ(hm)corresponding to the critical frequencies 2 : 1, 2 : 2 and 2 : 3 are situated in the interior of the


interval (0, 1), i.e. 0 < κ(hm) < Jra < 1. Since the maps g2−k,−2 for k = 1, 2, 3, . . . may well be of

the same size as g20,0, it generally will not be possible to establish passage through resonance for allsolutions.

4.58 c. ≈ 5 ≤ a For these values of a, all resonances are ”very close” to κ = 1 (cf. figure 4.19 where agraph of Ω together with the resonant values κkm are plotted for a = 1.1, 2.1, 4.1, 8.1). On a largedomain of the phase space it generally suffices to discuss the ”drift” in the outer zones given byg20,0.

2:1

2:42:3

2:2

κ

Ω(κ)

1

2:8

Figure 4.19: Graph of Ω(κ) for various choices of a

4.4.4 On the Capture in Resonance

Recalling the results found in section 2.3.4 we note that it is possible that some solutions of the reducedsystem (2.1) are captured near a resonance hm, i.e. satisfy |h(t)− hm| ≤ b0 |ε| for all t ≥ t0 (for someconstants b0, t0, see also (2.28)). By consequence of definition 1.5.1 the set h = hm in the (ϕ, h)–spacecorresponds to the trajectory of the solution (q, p)(t; 0,P(hm)) of the Hamiltonian system (1.2) withinitial value (0,P(hm)) and thus to the level curve Lhm of H through (0,P(hm)) in the (Q,P )–space.

As the transformations (1.15), (1.86) from (q, p)–coordinates into (Q,P )–coordinates are near–identical(up to O(ε2)–terms), we conclude that in the case of a capture near the resonance h = hm the corre-sponding solution (q, p)(t) satisfies

dist (Lhm , (q, p)(t)) = O(ε), ∀ t ≥ t0.

For large values of |P(hm)|, i.e. in the far upper or lower domain, the level curves Lhm are close to thelines p = P(hm). In view of (4.1.3) a captured solution then is of the form

ϑ(τ(t)) = (P(hm) + 1) t+ ε α(t, ε)

where α is bounded. If (ϕ, h)(t) is attracted by a λ 2π–periodic solution near h = hm (λ ∈ R∗+), then the

map α approaches a λ 2π–periodic function as t→ ∞.

We finally note that the values P(hm) are obtained from the values κ(hm) = κkm via the identities

P(hm) =

a/κkm in the upper domain

a κkm in the central domain

−a/κkm in the lower domain.


4.4.5 Finite Convolution via Discrete Fourier Transformation

We refer the reader to [16], pages 111 ff.

calcM := proc(m) 2^(m-2)-1 end

My2Maple := proc(alpha)

local Re_a,Im_a,k;global M;

Re_a := table([seq(k=evalc(Re(alpha[k-M])),k = 0 .. 2*M),seq(k = 0,k = 2*M+1 .. 4*M+3)]);Im_a := table([seq(k=evalc(Im(alpha[k-M])),k = 0 .. 2*M),seq(k = 0,k = 2*M+1 .. 4*M+3)]);[array([seq(Re_a[k],k = 0 .. 4*M+3)]),array([seq(Im_a[k],k = 0 .. 4*M+3)])]

end

LinConv := proc(Re_a, Im_a, Re_b, Im_b)

local k, DFT_a, DFT_b, Re_DFT_a, Im_DFT_a, Re_DFT_b, Im_DFT_b,Re_ab, Im_ab;global M, mm;

Re_DFT_a := copy(Re_a);Im_DFT_a := copy(Im_a);Re_DFT_b := copy(Re_b);Im_DFT_b := copy(Im_b);evalhf(FFT(mm, var(Re_DFT_a), var(Im_DFT_a)));evalhf(FFT(mm, var(Re_DFT_b), var(Im_DFT_b)));Re_ab := array([seq( Re_DFT_a[k]*Re_DFT_b[k] - Im_DFT_a[k]*Im_DFT_b[k],k=1 .. 4*M + 4)]);Im_ab := array([seq( Re_DFT_a[k]*Im_DFT_b[k] + Im_DFT_a[k]*Re_DFT_b[k],k=1 .. 4*M + 4)]);evalhf(iFFT(mm, var(Re_ab), var(Im_ab)));[array([seq(Re_ab[k], k = 1 .. 4*M + 4)]), array([seq(Im_ab[k], k = 1 .. 4*M + 4)])]

end

shift2M := proc(Re_a,Im_a)

local k;global M,cutoff;

for k from -M to M doif abs(Re_a[2*M+k+1]) < cutoff then Re_a[2*M+k+1] := 0 fi;if abs(Im_a[2*M+k+1]) < cutoff then Im_a[2*M+k+1] := 0 fi

od;table([seq(k = Re_a[2*M+k+1]+I*Im_a[2*M+k+1],k = -M .. M)])

end

ComplexFold := proc(alpha1,alpha2)local k,a,b,ab;

a := My2Maple(alpha1);b := My2Maple(alpha2);ab := LinConv(a[1],a[2],b[1],b[2]);ab := shift2M(ab[1],ab[2])

end


4.4.6 Additional Programmcode

For the interested reader we provide here the remaining code which was used to generate the outputpresented in section 4.5.

MakeG00datas := proc(mm, grid, kappa1, kappa2)

local i, kappa, G00, G00val, dn, iksn, cn, u, a, b, alpha,alphacc, beta, betacc, alphabeta, alphabetacc, abeta, bbeta,abetacc, bbetacc, S, starti, endi;global apar, zeta, z, rho, m, M, v, vb;

G00 := array(0 .. grid);printf(‘******************* Calculation of G_0,0(kappa) ***********************\n‘);printf(‘--> parameter a=%g rho=%g m=%g\n‘, apar, rho, m);printf(‘--> kappa in [%g, %g]\n‘, kappa1, kappa2);printf(‘--> kappa-steps = %g \n‘, (kappa2 - kappa1)/grid);printf(‘(4*M is the size of arrays used during Discrete Fourier Transformation)\n‘);printf(‘--------------------------------------------------------------------------\n‘);printf(‘| M | kappa | G_0,0(kappa)/(a*kappa) | G_0,0(kappa)/(a*kappa)^2 |‘);print();M := calcM(mm);if kappa1 = 0 then starti := 1 else starti := 0 fi;if kappa2 = 1 then endi := grid - 1 else endi := grid fi;for i from starti to endi do

kappa := kappa1 + i*(kappa2 - kappa1)/grid;dn := dn_build(kappa);iksn := iksn_build(kappa);cn := cn_build(kappa);a := a_build(dn, iksn);b := b_build(kappa, cn);alpha := alpha_build(a);beta := beta_build(iksn, dn, b);alphabeta := alphabeta_build(alpha, beta);abeta := abeta_build(a, beta);bbeta := bbeta_build(b, beta);alphacc := alphacc_build(alpha);alphabetacc := alphabetacc_build(alphabeta);abetacc := abetacc_build(abeta);bbetacc := bbetacc_build(bbeta);S := S1_build(kappa, alpha, alphacc);G00val := evalG00(beta,alphabeta,abeta,bbeta,alphabetacc,abetacc,bbetacc,zeta,z,S);G00[i] := [kappa, G00val/(apar*kappa)];printf(‘| %2.0f | %1.10f | %+1.10f | %+1.10f |‘,

M, kappa, G00val/(apar*kappa), G00val/(apar^2*kappa^2));print()

od;printf(‘******************** CALCULATION COMPLETE ******************\n‘);[seq(G00[i], i = starti .. endi)]

end


MakeGk2datas := proc(k, mm, grid, kappa1, kappa2, Resh)

local i, starti, endi, kappa, Gk2, Gk2val, G00val, bbeta, bbetacc,dn, iksn, cn, a, b, alpha, alphacc, beta, alphabeta, alphabetacc,abeta, abetacc, S, passStr, mmm;

global apar, zeta, z, M, v, vb;

Gk2 := array(0 .. grid);lprint();printf(‘\n**************** Calculation of 2*|G_-k,2(kappa)| *****************‘);lprint();printf(‘--> parameter a=%g rho=%g m=%g\n‘, apar, rho, m);printf(‘--> 2 : %g Resonance in %g \n‘, k, Resh);printf(‘--> kappa in [%g, %g]\n‘, kappa1, kappa2);printf(‘--> kappa-steps = %g‘, (kappa2 - kappa1)/grid);lprint();printf(‘(4*M is the size of arrays used during Discrete Fourier Transformation)\n‘);printf(‘-----------------------------------------------------------------------\n‘);printf(‘| M | kappa | G_%g,2(kappa)/(a*kappa) | G_0,0(kappa)/(a*kappa)

| passage |‘, -k);print();if kappa1 = 0 then starti := 1 else starti := 0 fi;if kappa2 = 1 then endi := grid - 1 else endi := grid fi;for i from starti to endi do

kappa := kappa1 + i*(kappa2 - kappa1)/grid;M := calcM(mm);mmm := mm;while evalb(M < abs(k)) do

mmm := mmm + 1; M := calcM(mmm)od;dn := dn_build(kappa);iksn := iksn_build(kappa);cn := cn_build(kappa);a := a_build(dn, iksn);b := b_build(kappa, cn);alpha := alpha_build(a);beta := beta_build(iksn, dn, b);alphabeta := alphabeta_build(alpha, beta);abeta := abeta_build(a, beta);bbeta := bbeta_build(b, beta);alphacc := alphacc_build(alpha);alphabetacc := alphabetacc_build(alphabeta);abetacc := abetacc_build(abeta);bbetacc := bbetacc_build(bbeta);S := S1_build(kappa, alpha, alphacc);Gk2val := evalGk2(k, beta, alphabeta, abeta, alphabetacc, abetacc, zeta, z, S);Gk2[i] := [kappa, Gk2val/(apar*kappa)];G00val := evalG00(beta,alphabeta,abeta,bbeta,alphabetacc,abetacc,bbetacc,zeta,z,S);if abs(G00val) > Gk2val then passStr := ‘ certain ‘else passStr := ‘ mostly ‘fi;printf(‘| %2.0f | %1.10f | %+1.10f | %+1.10f |%s|‘,

M, kappa, Gk2val/(apar*kappa), G00val/(apar*kappa), passStr);print()

od;printf(‘******************** CALCULATION COMPLETE *************************\n‘);[seq(Gk2[i], i = starti .. endi)]

end


RunIt := proc(kappa1, kappa2, grid, Resgrid, mm, delta, CalculateResonances)

local ResonanceList, k, j, G00Plot, G00Plotdata, Gk2Plot,Gk2Plotdata, Plot00Title, ommin, kmin, kmax, Plotk2Title,Filek2Name, kap1, kap2, ResNr, Resh, Output;

global apar, rho, m, M, zeta, z;

M := calcM(mm);Plot00Title := cat(‘G_0,0(kappa) ( abar=‘,convert(apar, name), ‘ , rho=‘,

convert(evalf(rho, 3), name), ‘ , m=‘, convert(m, name),‘ , M=‘,convert(calcM(mm), name), ‘)‘);

G00Plotdata := MakeG00datas(mm, grid, kappa1, kappa2);G00Plot := [Plot00Title, G00Plotdata];if evalb(CalculateResonances) then

ommin := Omega(kappa2, apar);kmin := trunc(6/ommin) + 1;kmax := min(kmin, 10);ResonanceList := DetectResonances(kmax, kappa1, kappa2);if 0 < ResonanceList[1] then

Gk2Plot := array(1 .. ResonanceList[1]);Gk2Plotdata := array(1 .. ResonanceList[1]);for k to ResonanceList[1] do

ResNr := abs(ResonanceList[2][k][1]);Resh := abs(ResonanceList[2][k][2]);kap1 := Resh - delta;kap2 := Resh + delta;if kap1 < 0 then kap1 := Resh fi;if 1 < kap2 then kap2 := Resh fi;Gk2Plotdata[k] := MakeGk2datas(ResNr, mm, Resgrid, kap1, kap2, Resh);Plotk2Title := cat(‘G_-‘, convert(ResNr, name), ‘,

2(kappa) ( abar=‘,convert(apar, name), ‘ , rho=‘, convert(evalf(rho, 3), name), ‘m=‘,convert(m, name), ‘M=‘, convert(calcM(mm), name), ‘)‘);

Gk2Plot[k] := [Plotk2Title, Gk2Plotdata[k]]od

fielse ResonanceList := [0]fi;[ResonanceList[1] + 2, G00Plot, seq(Gk2Plot[j], j = 1 .. ResonanceList[1])]

end


4.5 Numerical Evaluations, Discussion Following Chapter 2

In this section we present the results found via numerical evaluation of the formulae defining the mapsG0,0 and G−k,−2 given by (4.57) together with (4.46), (4.47) in section 4.3.5. The calculations werecarried out for choices of a ∈ 0.54, 4.1, 20.38, ,m ∈ 0, 1. Each subsection to follow corresponds to aparticular set of parameters. The output of the evaluations is organized as follows:

• Graphs ofG0,0(κ)a/κ on the upper,

G0,0(κ)a κ on the central and

G0,0(κ)−a/κ on the lower domains.

Recall that as κ → 1 points approach a small ε–independent neighbourhood of the seperatrix. Inthe central domain (where P(h) = a κ(h)) points near the periodic solution correspond to κ → 0while for the upper (P(h) = a/κ(h)) and lower (P(h) = −a/κ(h)) domain points tend towards ±∞as κ→ 0.

• Alphanumerical output of the calculation routines. For each of the upper, central and thelower domain the output of the calculation scheme is given. It is divided into three parts:

– Calculation of G0,0(κ) : The output includes a header showing further specification of the

parameters followed by a listing of the valuesG0,0(κ)a κ ,

(

±G0,0(κ)a/κ

)

andG0,0(κ)

(a κ)2,(

G0,0(κ)

(a/κ)2

)

eval-

uated at 25 equidistant points in the interval [0, 1).

By looking at the values found forG0,0(κ)

(a κ)2near κ = 0 in the central domain, an approximation

for the value g2,10,0 is found (cf. section 4.4.2). The reader is invited to compare these results tothe graph depicted in figure 4.36.

– Detection of resonances: The output of the routine detecting resonances contains the list ofthe ratio of the resonances found together with the corresponding value κ(hm). The maximumorder ′′kmax′′ of resonances to be considered is set to 2 : 10 here. If no resonances were foundon the domain considered for κ the output consist of a header row and a footer row only.

– Calculation(s) of G−k,−2 : For each resonance listed the valuesG−k,−2(κ)

aκ ,(

±G−k,−2(κ)a/κ

)

are

evaluated and compared withG0,0(κ)aκ ,

(

±G0,0(κ)a/κ

)

. In view of the discussion in section 4.4.2

this makes it possible to decide whether a passage through the resonance is certain for allsolutions or for most solutions (i.e. up to a set of size O(ε), cf. section 2.3.4). This is indicatedin the last column with the marks certain or mostly.

• Schematic Phase Portrait A schematic sketch of the phase portrait of the reduced systemshowing the average drifts (arrows) and attracting sets (light grey, if existing) closes each subsection.Moreover resonance curves a shown (dashed) if the passage through resonance may be establishedfor most solutions, i.e. up to a set of order O(ε).

4.5. Numerical Evaluations, Discussion Following Chapter 2 173

4.5.1 a = 0.54, = 0, m = 0

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.2 0.4 0.6 0.8 1

G_0,0(kappa) ( abar=.54 , rho=0 , m=0 , M=63)

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0.2 0.4 0.6 0.8 1


0

0.2

0.4

0.6

0.8

0.2 0.4 0.6 0.8 1


Figure 4.20: a = 0.54, = 0, m = 0 : plot ofG0,0(κ)a/κ on the upper,

G0,0(κ)aκ on the central and

G0,0(κ)−a/κ

on the lower domains


upper domain :

*********************** Calculation of G_0,0(kappa) ***************************

--> parameter a=.54 rho=0 m=0

--> kappa in [0, .99]

--> kappa-steps = .0396

(4*M is the size of arrays used during Discrete Fourier Transformation)

---------------------------------------------------------------------------------

| M | kappa | G_0,0(kappa)/(a/kappa) | G_0,0(kappa)/(a/kappa)^2 |

| 63 | .0396000000 | -.0686447851 | -0.0050339509 |

| 63 | .0792000000 | -.1300828309 | -0.0190788152 |

| 63 | .1188000000 | -.1866895908 | -0.0410717099 |

| 63 | .1584000000 | -.2400945790 | -0.0704277431 |

| 63 | .1980000000 | -.2914430218 | -0.1068624413 |

| 63 | .2376000000 | -.3415132761 | -0.1502658415 |

| 63 | .2772000000 | -.3907754230 | -0.2005980505 |

| 63 | .3168000000 | -.4394220391 | -0.2577942629 |

| 63 | .3564000000 | -.4873850566 | -0.3216741374 |

| 63 | .3960000000 | -.5343471041 | -0.3918545430 |

| 63 | .4356000000 | -.5797537987 | -0.4676680642 |

| 63 | .4752000000 | -.6228324110 | -0.5480925217 |

| 63 | .5148000000 | -.6626210395 | -0.6316987243 |

| 63 | .5544000000 | -.6980104038 | -0.7166240146 |

| 63 | .5940000000 | -.7277974411 | -0.8005771852 |

| 63 | .6336000000 | -.7507462550 | -0.8808756058 |

| 63 | .6732000000 | -.7656480610 | -0.9545079160 |

| 63 | .7128000000 | -.7713680246 | -1.0182057925 |

| 63 | .7524000000 | -.7668630276 | -1.0684958185 |

| 63 | .7920000000 | -.7511480530 | -1.1016838111 |

| 63 | .8316000000 | -.7231714910 | -1.1136840961 |

| 63 | .8712000000 | -.6815005426 | -1.0994875420 |

| 63 | .9108000000 | -.6234780025 | -1.0515995642 |

| 63 | .9504000000 | -.5421347699 | -0.9541571951 |

| 63 | .9900000000 | -.3983523535 | -0.7303126481 |

******************** CALCULATION COMPLETE *************************

******************* Detection of Resonances ***********************

2 : 1-Resonance in +.1343870049......

2 : 2-Resonance in +.2651460995......

2 : 3-Resonance in +.3888961596......

2 : 4-Resonance in +.5027329013......

******************* DETECTION COMPLETE ***********************

*********************** Calculation of G_-k,2(kappa)| ***************************


--> 2 : 1 Resonance in .134387

--> kappa in [.084387, .184387]



-----------------------------------------------------------------------------------------

| M | kappa | G_-1,2(kappa)/(a/kappa) | G_0,0(kappa)/(a/kappa) | passage |

| 63 | .0843870049 | +0.0000000000 | -.1377315714 | certain |

| 63 | .1343870049 | +0.0000000000 | -.2080198644 | certain |

| 63 | .1843870049 | +0.0000000000 | -.2739679876 | certain |

******************************* CALCULATION COMPLETE **********************************




--> 2 : 2 Resonance in .265146

--> kappa in [.215146, .315146]



-----------------------------------------------------------------------------------------


| 63 | .2151460995 | +.4645644541 | -.3132452504 | mostly |

| 63 | .2651460995 | +.4819943004 | -.3758492763 | mostly |

| 63 | .3151460995 | +.4966052351 | -.4374026487 | mostly |




--> 2 : 3 Resonance in .388896

--> kappa in [.338896, .438896]



-----------------------------------------------------------------------------------------


| 63 | .3388961596 | +0.0000000000 | -.4662836238 | certain |

| 63 | .3888961596 | +0.0000000000 | -.5260180190 | certain |

| 63 | .4388961596 | +0.0000000000 | -.5834404315 | certain |




--> 2 : 4 Resonance in .502732

--> kappa in [.452732, .552732]



-----------------------------------------------------------------------------------------


| 63 | .4527329013 | +.2001595457 | -.5987298864 | certain |

| 63 | .5027329013 | +.2280917206 | -.6509090303 | certain |

| 63 | .5527329013 | +.2544413511 | -.6966249001 | certain |


As seen in the plot on top of figure 4.20 the drift G0,0(κ) is negative for all values κ ∈ (0, 0.99] evaluatednumerically. Hence away from resonances (i.e. in the outer zone) the solutions of the correspondingreduced system tend towards h = −∞ i.e. κ → 1 thus towards an ε–independant small neighbourhoodof the upper separatrix.The alphanumerical output indicates that up to an O(ε)–set (cf. proposition 2.3.11) all solutions passthrough the 2 : 2 resonance arising at κkm ≈ 0.26 as |G0,0(0.26)| < G−2,2(0.26). Hence for at most anO(ε)–set of solutions a capture in this 2 : 2 resonance is possible. The value κkm ≈ 0.26 corresponds toP(hm) = a/κkm ≈ 2 (cf. section 4.4.4).

The remaining resonances found are passed by all solutions, as it was discussed in lemma 2.3.4.


central domain :

******************* Calculation of G_0,0(kappa) ***********************


--> kappa in [0, .99]



--------------------------------------------------------------------------------

| M | kappa | G_0,0(kappa)/(a*kappa) | G_0,0(kappa)/(a*kappa)^2 |

| 63 | .0396000000 | -.0101616353 | -.4751980619 |

| 63 | .0792000000 | -.0202893535 | -.4744050119 |

| 63 | .1188000000 | -.0303493075 | -.4730843549 |

| 63 | .1584000000 | -.0403077848 | -.4712376638 |

| 63 | .1980000000 | -.0501312599 | -.4688670030 |

| 63 | .2376000000 | -.0597864278 | -.4659747772 |

| 63 | .2772000000 | -.0692402077 | -.4625635173 |

| 63 | .3168000000 | -.0784597068 | -.4586355855 |

| 63 | .3564000000 | -.0874121242 | -.4541927725 |

| 63 | .3960000000 | -.0960645724 | -.4492357485 |

| 63 | .4356000000 | -.1043837815 | -.4437633130 |

| 63 | .4752000000 | -.1123356327 | -.4377713583 |

| 63 | .5148000000 | -.1198844445 | -.4312514191 |

| 63 | .5544000000 | -.1269918882 | -.4241886064 |

| 63 | .5940000000 | -.1336153361 | -.4165585988 |

| 63 | .6336000000 | -.1397053106 | -.4083231349 |

| 63 | .6732000000 | -.1452014534 | -.3994230249 |

| 63 | .7128000000 | -.1500259297 | -.3897668292 |

| 63 | .7524000000 | -.1540721083 | -.3792114821 |

| 63 | .7920000000 | -.1571838147 | -.3675266900 |

| 63 | .8316000000 | -.1591137393 | -.3543230793 |

| 63 | .8712000000 | -.1594287613 | -.3388871062 |

| 63 | .9108000000 | -.1572490262 | -.3197210150 |

| 63 | .9504000000 | -.1502475728 | -.2927569928 |

| 63 | .9900000000 | -.1254297127 | -.2346234806 |



2 : 8-Resonance in +.5191173717......

2 : 9-Resonance in +.7483112555......

2 : 10-Resonance in +.8538470933......


******************* Calculation of G_-k,2(kappa)| ***********************


--> 2 : 8 Resonance in .519117

--> kappa in [.469117, .569117]



-----------------------------------------------------------------------------------------

| M | kappa | G_-8,2(kappa)/(a*kappa) | G_0,0(kappa)/(a*kappa) | passage |

| 63 | .4691173717 | +.0000757834 | -.1111394979 | certain |

| 63 | .5191173717 | +.0001628013 | -.1206814887 | certain |

| 63 | .5691173717 | +.0003304881 | -.1295124702 | certain |





--> 2 : 9 Resonance in .748311

--> kappa in [.698311, .798311]



-----------------------------------------------------------------------------------------


| 63 | .6983112555 | +.0005194597 | -.1483446195 | certain |

| 63 | .7483112555 | +.0010457119 | -.1536943953 | certain |

| 63 | .7983112555 | +.0020912121 | -.1575788339 | certain |




--> 2 : 10 Resonance in .853847

--> kappa in [.803847, .903847]



-----------------------------------------------------------------------------------------


| 63 | .8038470933 | +.0008016925 | -.1578998734 | certain |

| 63 | .8538470933 | +.0018221840 | -.1595301896 | certain |

| 63 | .9038470933 | +.0043835408 | -.1578798979 | certain |


As the drift G0,0(κ) is negative for all values κ ∈ (0, 0.99], h in the averaged reduced system is negative(for h > 0). Hence the solutions tend towards the invariant set κ = 0, h = 0, respectively i.e. the periodicsolution at the origin. The solutions may not be captured in the resonances arising. Together with theresults found in section 4.6 to follow, we will see that the periodic solution h = 0 is globally attractive

and stable in the entire central domain. From the last column showingG0,0(κ)

(κ a)2we estimate g2,10,0 ≈ −0.47

which coincides with the value presented in figure 4.36.


lower domain :

************************** Calculation of G_0,0(kappa) ***************************


--> kappa in [0, .99]



----------------------------------------------------------------------------------

| M | kappa | G_0,0(kappa)/(-a/kappa) | G_0,0(kappa)/(-a/kappa)^2 |

| 63 | .0396000000 | +.0796445290 | -.0058405987 |

| 63 | .0792000000 | +.1774760156 | -.0260298156 |

| 63 | .1188000000 | +.3105204651 | -.0683145023 |

| 63 | .1584000000 | +.5257107500 | -.1542084866 |

| 63 | .1980000000 | +.8767114389 | -.3214608609 |

| 63 | .2376000000 | +.7773337592 | -.3420268540 |

| 63 | .2772000000 | +.4115375150 | -.2112559244 |

| 63 | .3168000000 | +.2727148538 | -.1599927142 |

| 63 | .3564000000 | +.1994313322 | -.1316246793 |

| 63 | .3960000000 | +.1390129921 | -.1019428609 |

| 63 | .4356000000 | +.0832007341 | -.0671152589 |

| 63 | .4752000000 | +.0325510962 | -.0286449646 |

| 63 | .5148000000 | -.0113934889 | +.0108617928 |

| 63 | .5544000000 | -.0481280794 | +.0494114948 |

| 63 | .5940000000 | -.0781084755 | +.0859193230 |

| 63 | .6336000000 | -.1021867688 | +.1198991421 |

| 63 | .6732000000 | -.1212749731 | +.1511894665 |

| 63 | .7128000000 | -.1361830077 | +.1797615702 |

| 63 | .7524000000 | -.1475541227 | +.2055920776 |

| 63 | .7920000000 | -.1558358983 | +.2285593176 |

| 63 | .8316000000 | -.1612471207 | +.2483205659 |

| 63 | .8712000000 | -.1636952556 | +.2640950124 |

| 63 | .9108000000 | -.1625202959 | +.2741175659 |

| 63 | .9504000000 | -.1554663079 | +.2736207019 |

| 63 | .9900000000 | -.1291750079 | +.2368208479 |



2 : 1-Resonance in +.1343870049......

2 : 2-Resonance in +.2651460995......

2 : 3-Resonance in +.3888961596......

2 : 4-Resonance in +.5027329013......




--> 2 : 1 Resonance in .134387

--> kappa in [.084387, .184387]



-------------------------------------------------------------------------------------------

| M | kappa | G_-1,2(kappa)/(-a/kappa) | G_0,0(kappa)/(-a/kappa) | passage |

| 63 | .0843870049 | +0.0000000000 | +.1923262653 | certain |

| 63 | .1343870049 | +0.0000000000 | +.3810181802 | certain |

| 63 | .1843870049 | +0.0000000000 | +.7503725033 | certain |





--> 2 : 2 Resonance in .265146

--> kappa in [.215146, .315146]



-------------------------------------------------------------------------------------------


| 63 | .2151460995 | -.0318656728 | +.9486125303 | certain |

| 63 | .2651460995 | -.0348682456 | +.4909921780 | certain |

| 63 | .3151460995 | -.0384116075 | +.2765091762 | certain |




--> 2 : 3 Resonance in .388896

--> kappa in [.338896, .438896]



-------------------------------------------------------------------------------------------


| 63 | .3388961596 | +0.0000000000 | +.2289045592 | certain |

| 63 | .3888961596 | +0.0000000000 | +.1494734630 | certain |

| 63 | .4388961596 | +0.0000000000 | +.0787643246 | certain |




--> 2 : 4 Resonance in .502732

--> kappa in [.452732, .552732]



-------------------------------------------------------------------------------------------


| 63 | .4527329013 | -.0017777368 | +.0605512470 | certain |

| 63 | .5027329013 | -.0023731029 | +.0012323718 | mostly |

| 63 | .5527329013 | -.0031129066 | -.0467229812 | certain |


For κ ∈ (0,≈ 0.5) the drift G0,0(κ)/(−a/κ) and thus h is positive,

P

P=-1

Q

Figure 4.21: a = 0.54, = 0,m = 0

while κ ∈ (≈ 0.5, 0.99) implies h < 0. Thus the solutions in thelower domain tend towards the set κ ≈ 0.5 which is equivalent toP ≈ −a/κ ≈ −1. This attractive set contains solutions of the formϑ(τ(t)) ≈ ε α(t, ε) (cf. the formula found in section 4.4.4) which maychange the sign. From a physical point of view this corresponds to anoscillation of the rotor. The 2 : 4 resonance detected in κkm ≈ 0.502is very close to the zero of the function G0,0. As this zero and theresonance are possibly identical, the results of section 2.3 might notbe applicable here.

For a discussion of the separatrix region (i.e. the white region in fig-ure 4.21) we refer the reader to section 4.7.


4.5.2 a = 0.54, = 1, m = 0

-14

-12

-10

-8

-6

-4

-2

0.2 0.4 0.6 0.8 1

G_0,0(kappa) ( abar=.54 , rho=1.0 , m=0 , M=63)

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0.2 0.4 0.6 0.8 1


0

2

4

6

8

10

12

0.2 0.4 0.6 0.8 1




G0,0(κ)−a/κ



upper domain :



--> kappa in [0, .99]



--------------------------------------------------------------------------------


| 63 | .0396000000 | -14.1941182078 | -1.0409020019 |

| 63 | .0792000000 | -7.4260782072 | -1.0891581370 |

| 63 | .1188000000 | -5.1982391236 | -1.1436126072 |

| 63 | .1584000000 | -4.1031203872 | -1.2035819802 |

| 63 | .1980000000 | -3.4600266337 | -1.2686764323 |

| 63 | .2376000000 | -3.0424373556 | -1.3386724364 |

| 63 | .2772000000 | -2.7533895421 | -1.4134066316 |

| 63 | .3168000000 | -2.5443388122 | -1.4926787698 |

| 63 | .3564000000 | -2.3881189587 | -1.5761585127 |

| 63 | .3960000000 | -2.2681294223 | -1.6632949097 |

| 63 | .4356000000 | -2.1734264733 | -1.7532306884 |

| 63 | .4752000000 | -2.0962798153 | -1.8447262375 |

| 63 | .5148000000 | -2.0308740516 | -1.9360999292 |

| 63 | .5544000000 | -1.9725890858 | -2.0251914614 |

| 63 | .5940000000 | -1.9175930801 | -2.1093523881 |

| 63 | .6336000000 | -1.8626099345 | -2.1854623231 |

| 63 | .6732000000 | -1.8047808130 | -2.2499600802 |

| 63 | .7128000000 | -1.7415649398 | -2.2988657206 |

| 63 | .7524000000 | -1.6706337731 | -2.3277497239 |

| 63 | .7920000000 | -1.5897068156 | -2.3315699962 |

| 63 | .8316000000 | -1.4962428055 | -2.3042139205 |

| 63 | .8712000000 | -1.3867758994 | -2.2373317843 |

| 63 | .9108000000 | -1.2551877767 | -2.1170833834 |

| 63 | .9504000000 | -1.0863723838 | -1.9120153956 |

| 63 | .9900000000 | -.7994679938 | -1.4656913221 |



2 : 1-Resonance in +.1343870049......

2 : 2-Resonance in +.2651460995......

2 : 3-Resonance in +.3888961596......

2 : 4-Resonance in +.5027329013......




--> 2 : 1 Resonance in .134387

--> kappa in [.084387, .184387]



-----------------------------------------------------------------------------------------


| 63 | .0843870049 | +0.0000000000 | -7.0131241237 | certain |

| 63 | .1343870049 | +0.0000000000 | -4.6876280257 | certain |

| 63 | .1843870049 | +0.0000000000 | -3.6482952790 | certain |





--> 2 : 2 Resonance in .265146

--> kappa in [.215146, .315146]



-----------------------------------------------------------------------------------------


| 63 | .2151460995 | +.4645644541 | -3.2588621405 | certain |

| 63 | .2651460995 | +.4819943004 | -2.8312318725 | certain |

| 63 | .3151460995 | +.4966052351 | -2.5518692869 | certain |




--> 2 : 3 Resonance in .388896

--> kappa in [.338896, .438896]



-----------------------------------------------------------------------------------------


| 63 | .3388961596 | +0.0000000000 | -2.4518881209 | certain |

| 63 | .3888961596 | +0.0000000000 | -2.2875287270 | certain |

| 63 | .4388961596 | +0.0000000000 | -2.1664254389 | certain |




--> 2 : 4 Resonance in .502732

--> kappa in [.452732, .552732]



-----------------------------------------------------------------------------------------


| 63 | .4527329013 | +.2001595457 | -2.1382691569 | certain |

| 63 | .5027329013 | +.2280917206 | -2.0498557721 | certain |

| 63 | .5527329013 | +.2544413511 | -1.9749517859 | certain |


central domain :



--> kappa in [0, .99]



--------------------------------------------------------------------------------


| 63 | .0396000000 | -.0208515378 | -.9750999750 |

| 63 | .0792000000 | -.0416565339 | -.9740117374 |

| 63 | .1188000000 | -.0623683167 | -.9721959833 |

| 63 | .1584000000 | -.0829399403 | -.9696495088 |

| 63 | .1980000000 | -.1033240111 | -.9663674819 |

| 63 | .2376000000 | -.1234724649 | -.9623430677 |

| 63 | .2772000000 | -.1433362740 | -.9575668995 |

| 63 | .3168000000 | -.1628650526 | -.9520263552 |

| 63 | .3564000000 | -.1820065203 | -.9457045784 |

| 63 | .3960000000 | -.2007057664 | -.9385791548 |

| 63 | .4356000000 | -.2189042325 | -.9306203130 |

| 63 | .4752000000 | -.2365382917 | -.9217884546 |

| 63 | .5148000000 | -.2535372413 | -.9120307106 |

| 63 | .5544000000 | -.2698204188 | -.9012760503 |

| 63 | .5940000000 | -.2852929789 | -.8894281672 |


| 63 | .6336000000 | -.2998395469 | -.8763548299 |

| 63 | .6732000000 | -.3133143740 | -.8618713662 |

| 63 | .7128000000 | -.3255254323 | -.8457139094 |

| 63 | .7524000000 | -.3362073506 | -.8274936269 |

| 63 | .7920000000 | -.3449721435 | -.8066127562 |

| 63 | .8316000000 | -.3512110287 | -.7820957118 |

| 63 | .8712000000 | -.3538724650 | -.7522031447 |

| 63 | .9108000000 | -.3508556121 | -.7133647507 |

| 63 | .9504000000 | -.3367092493 | -.6560770695 |

| 63 | .9900000000 | -.2817046189 | -.5269446670 |



2 : 8-Resonance in +.5191173717......

2 : 9-Resonance in +.7483112555......

2 : 10-Resonance in +.8538470933......




--> 2 : 8 Resonance in .519117

--> kappa in [.469117, .569117]



-----------------------------------------------------------------------------------------


| 63 | .4691173717 | +.0000757834 | -.2338690892 | certain |

| 63 | .5191173717 | +.0001628013 | -.2553488358 | certain |

| 63 | .5691173717 | +.0003304881 | -.2756710900 | certain |




--> 2 : 9 Resonance in .748311

--> kappa in [.698311, .798311]



-----------------------------------------------------------------------------------------


| 63 | .6983112555 | +.0005194597 | -.3212190507 | certain |

| 63 | .7483112555 | +.0010457119 | -.3351852498 | certain |

| 63 | .7983112555 | +.0020912121 | -.3461559201 | certain |




--> 2 : 10 Resonance in .853847

--> kappa in [.803847, .903847]



-----------------------------------------------------------------------------------------


| 63 | .8038470933 | +.0008016925 | -.3471396203 | certain |

| 63 | .8538470933 | +.0018221840 | -.3532445112 | certain |

| 63 | .9038470933 | +.0043835408 | -.3519559769 | certain |



lower domain :



--> kappa in [0, .99]



----------------------------------------------------------------------------------


| 63 | .0396000000 | +13.2055101840 | -.9684040801 |

| 63 | .0792000000 | +6.4750426365 | -.9496729200 |

| 63 | .1188000000 | +4.3256140415 | -.9516350891 |

| 63 | .1584000000 | +3.3950590710 | -.9958839941 |

| 63 | .1980000000 | +3.0552187840 | -1.1202468874 |

| 63 | .2376000000 | +2.4926282995 | -1.0967564518 |

| 63 | .2772000000 | +1.7938434247 | -.9208396246 |

| 63 | .3168000000 | +1.4035556813 | -.8234193330 |

| 63 | .3564000000 | +1.1332772528 | -.7479629868 |

| 63 | .3960000000 | +.9141057892 | -.6703442454 |

| 63 | .4356000000 | +.7274598044 | -.5868175756 |

| 63 | .4752000000 | +.5670202826 | -.4989778487 |

| 63 | .5148000000 | +.4295771905 | -.4095302549 |

| 63 | .5544000000 | +.3122502478 | -.3205769211 |

| 63 | .5940000000 | +.2121129891 | -.2333242880 |

| 63 | .6336000000 | +.1264754786 | -.1483978949 |

| 63 | .6732000000 | +.0530401909 | -.0661234380 |

| 63 | .7128000000 | -.0100534027 | +.0132704915 |

| 63 | .7524000000 | -.0642396849 | +.0895072943 |

| 63 | .7920000000 | -.1105416397 | +.1621277382 |

| 63 | .8316000000 | -.1495505577 | +.2303078588 |

| 63 | .8712000000 | -.1813097835 | +.2925131175 |

| 63 | .9108000000 | -.2049101475 | +.3456151155 |

| 63 | .9504000000 | -.2168489325 | +.3816541212 |

| 63 | .9900000000 | -.1960317450 | +.3593915325 |



2 : 1-Resonance in +.1343870049......

2 : 2-Resonance in +.2651460995......

2 : 3-Resonance in +.3888961596......

2 : 4-Resonance in +.5027329013......




--> 2 : 1 Resonance in .134387

--> kappa in [.084387, .184387]



-------------------------------------------------------------------------------------------


| 63 | .0843870049 | +0.0000000000 | +6.0695030868 | certain |

| 63 | .1343870049 | +0.0000000000 | +3.8651670456 | certain |

| 63 | .1843870049 | +0.0000000000 | +3.1332914693 | certain |





--> 2 : 2 Resonance in .265146

--> kappa in [.215146, .315146]



-------------------------------------------------------------------------------------------


| 63 | .2151460995 | -.0318656728 | +2.9059731667 | certain |

| 63 | .2651460995 | -.0348682456 | +1.9643521831 | certain |

| 63 | .3151460995 | -.0384116075 | +1.4166207960 | certain |




--> 2 : 3 Resonance in .388896

--> kappa in [.338896, .438896]



---------------------------------------------------------------------------------_----------


| 63 | .3388961596 | +0.0000000000 | +1.2443227670 | certain |

| 63 | .3888961596 | +0.0000000000 | +.9507467398 | certain |

| 63 | .4388961596 | +0.0000000000 | +.7131588392 | certain |




--> 2 : 4 Resonance in .502732

--> kappa in [.452732, .552732]



-------------------------------------------------------------------------------------------


| 63 | .4527329013 | -.0017777368 | +.6550440373 | certain |

| 63 | .5027329013 | -.0023731029 | +.4691922593 | certain |

| 63 | .5527329013 | -.0031129066 | +.3168229222 | certain |


The qualitative behaviour is very similar to the preceeding situation

P

P=-1

Q

Figure 4.23: a = 0.54, = 1,m = 0

of section 4.5.1. Due to = 1 the drift remote from the separatrices,in the upper and lower domain becomes much larger than for = 0.As for the same reason capture in resonances (and in particular in the2 : 2 resonance at κkm ≈ 0.26 for the upper domain) does not appear.The attractive set in the lower domain moves to P ≈ −0.78. Thephysical interpretation then is different. As ϑ(τ(t)) is of the form0.22 t+ε α(t, ε), the rotor does not oscillate but rotate here. However,the mean angular speed of the rotor varies around 0.22ω (with respectto time τ) compared to the much larger angular speed ω of the periodicsolution h = 0 (or, equivalently (Q,P ) = (0, 0)). Hence for solutionsattracted in P ≈ −0.78 the synchronous motor rotates but with aslow speed that periodically increases and decreases.


4.5.3 a = 0.54, = 0, m = 1

-1.2

-1.1

-1

-0.9

-0.8

-0.7

-0.6

0.2 0.4 0.6 0.8 1

G_0,0(kappa) ( abar=.54 , rho=0 , m=1. , M=63)

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0.2 0.4 0.6 0.8 1


-0.4

-0.2

0

0.2

0.4

0.2 0.4 0.6 0.8 1




G0,0(κ)−a/κ



upper domain :



--> kappa in [0, .99]



--------------------------------------------------------------------------------


| 63 | .0396000000 | -.5684486690 | -.0416862357 |

| 63 | .0792000000 | -.6292972086 | -.0922969239 |

| 63 | .1188000000 | -.6849175690 | -.1506818651 |

| 63 | .1584000000 | -.7369333226 | -.2161671079 |

| 63 | .1980000000 | -.7864811552 | -.2883764235 |

| 63 | .2376000000 | -.8343280457 | -.3671043401 |

| 63 | .2772000000 | -.8809295277 | -.4522104909 |

| 63 | .3168000000 | -.9264600119 | -.5435232070 |

| 63 | .3564000000 | -.9708290474 | -.6407471713 |

| 63 | .3960000000 | -1.0136918647 | -.7433740341 |

| 63 | .4356000000 | -1.0544606008 | -.8505982180 |

| 63 | .4752000000 | -1.0923215199 | -.9612429375 |

| 63 | .5148000000 | -1.1262622058 | -1.0737033029 |

| 63 | .5544000000 | -1.1551105812 | -1.1859135300 |

| 63 | .5940000000 | -1.1775845283 | -1.2953429811 |

| 63 | .6336000000 | -1.1923469710 | -1.3990204459 |

| 63 | .6732000000 | -1.1980568549 | -1.4935775458 |

| 63 | .7128000000 | -1.1934016797 | -1.5752902172 |

| 63 | .7524000000 | -1.1770911815 | -1.6400803796 |

| 63 | .7920000000 | -1.1477803050 | -1.6834111140 |

| 63 | .8316000000 | -1.1038588667 | -1.6999426548 |

| 63 | .8712000000 | -1.0429454849 | -1.6826187157 |

| 63 | .9108000000 | -.9605278154 | -1.6200902486 |

| 63 | .9504000000 | -.8449448892 | -1.4871030049 |

| 63 | .9900000000 | -.6323385422 | -1.1592873274 |



2 : 1-Resonance in +.1343870049......

2 : 2-Resonance in +.2651460995......

2 : 3-Resonance in +.3888961596......

2 : 4-Resonance in +.5027329013......




--> 2 : 1 Resonance in .134387

--> kappa in [.084387, .184387]



-----------------------------------------------------------------------------------------


| 63 | .0843870049 | +0.0000000000 | -.6368394368 | certain |

| 63 | .1343870049 | +0.0000000000 | -.7057495124 | certain |

| 63 | .1843870049 | +0.0000000000 | -.7696721503 | certain |





--> 2 : 2 Resonance in .265146

--> kappa in [.215146, .315146]



-----------------------------------------------------------------------------------------


| 63 | .2151460995 | +.4645644541 | -.8073733772 | certain |

| 63 | .2651460995 | +.4819943004 | -.8668605718 | certain |

| 63 | .3151460995 | +.4966052351 | -.9245801579 | certain |




--> 2 : 3 Resonance in .388896

--> kappa in [.338896, .438896]



-----------------------------------------------------------------------------------------


| 63 | .3388961596 | +0.0000000000 | -.9513767685 | certain |

| 63 | .3888961596 | +0.0000000000 | -1.0061367346 | certain |

| 63 | .4388961596 | +0.0000000000 | -1.0577356779 | certain |




--> 2 : 4 Resonance in .502732

--> kappa in [.452732, .552732]



-----------------------------------------------------------------------------------------


| 63 | .4527329013 | +.2001595457 | -1.0712531265 | certain |

| 63 | .5027329013 | +.2280917206 | -1.1164024575 | certain |

| 63 | .5527329013 | +.2544413511 | -1.1540153912 | certain |


central domain :



--> kappa in [0, .99]



--------------------------------------------------------------------------------


| 63 | .0396000000 | -.0101616353 | -.4751980619 |

| 63 | .0792000000 | -.0202893535 | -.4744050119 |

| 63 | .1188000000 | -.0303493075 | -.4730843549 |

| 63 | .1584000000 | -.0403077848 | -.4712376638 |

| 63 | .1980000000 | -.0501312599 | -.4688670030 |

| 63 | .2376000000 | -.0597864278 | -.4659747772 |

| 63 | .2772000000 | -.0692402077 | -.4625635173 |

| 63 | .3168000000 | -.0784597068 | -.4586355855 |

| 63 | .3564000000 | -.0874121242 | -.4541927725 |

| 63 | .3960000000 | -.0960645724 | -.4492357485 |

| 63 | .4356000000 | -.1043837815 | -.4437633130 |

| 63 | .4752000000 | -.1123356327 | -.4377713583 |

| 63 | .5148000000 | -.1198844445 | -.4312514191 |

| 63 | .5544000000 | -.1269918882 | -.4241886064 |

| 63 | .5940000000 | -.1336153361 | -.4165585988 |


| 63 | .6336000000 | -.1397053106 | -.4083231349 |

| 63 | .6732000000 | -.1452014534 | -.3994230249 |

| 63 | .7128000000 | -.1500259297 | -.3897668292 |

| 63 | .7524000000 | -.1540721083 | -.3792114821 |

| 63 | .7920000000 | -.1571838147 | -.3675266900 |

| 63 | .8316000000 | -.1591137393 | -.3543230793 |

| 63 | .8712000000 | -.1594287613 | -.3388871062 |

| 63 | .9108000000 | -.1572490262 | -.3197210150 |

| 63 | .9504000000 | -.1502475728 | -.2927569928 |

| 63 | .9900000000 | -.1254297127 | -.2346234806 |



2 : 8-Resonance in +.5191173717......

2 : 9-Resonance in +.7483112555......

2 : 10-Resonance in +.8538470933......




--> 2 : 8 Resonance in .519117

--> kappa in [.469117, .569117]



-----------------------------------------------------------------------------------------


| 63 | .4691173717 | +.0000757834 | -.1111394979 | certain |

| 63 | .5191173717 | +.0001628013 | -.1206814887 | certain |

| 63 | .5691173717 | +.0003304881 | -.1295124702 | certain |




--> 2 : 9 Resonance in .748311

--> kappa in [.698311, .798311]



-----------------------------------------------------------------------------------------


| 63 | .6983112555 | +.0005194597 | -.1483446195 | certain |

| 63 | .7483112555 | +.0010457119 | -.1536943953 | certain |

| 63 | .7983112555 | +.0020912121 | -.1575788339 | certain |




--> 2 : 10 Resonance in .853847

--> kappa in [.803847, .903847]



-----------------------------------------------------------------------------------------


| 63 | .8038470933 | +.0008016925 | -.1578998734 | certain |

| 63 | .8538470933 | +.0018221840 | -.1595301896 | certain |

| 63 | .9038470933 | +.0043835408 | -.1578798979 | certain |



lower domain :



--> kappa in [0, .99]



----------------------------------------------------------------------------------


| 63 | .0396000000 | -.4201593548 | +.0308116860 |

| 63 | .0792000000 | -.3217383621 | +.0471882931 |

| 63 | .1188000000 | -.1877075130 | +.0412956528 |

| 63 | .1584000000 | +.0288720063 | -.0084691218 |

| 63 | .1980000000 | +.3816733055 | -.1399468786 |

| 63 | .2376000000 | +.2845189896 | -.1251883554 |

| 63 | .2772000000 | -.0786165896 | +.0403565160 |

| 63 | .3168000000 | -.2143231189 | +.1257362297 |

| 63 | .3564000000 | -.2840126584 | +.1874483545 |

| 63 | .3960000000 | -.3403317684 | +.2495766301 |

| 63 | .4356000000 | -.3915060679 | +.3158148948 |

| 63 | .4752000000 | -.4369380127 | +.3845054511 |

| 63 | .5148000000 | -.4750346552 | +.4528663713 |

| 63 | .5544000000 | -.5052282567 | +.5187010103 |

| 63 | .5940000000 | -.5278955627 | +.5806851189 |

| 63 | .6336000000 | -.5437874848 | +.6380439822 |

| 63 | .6732000000 | -.5536837671 | +.6902590963 |

| 63 | .7128000000 | -.5582166628 | +.7368459949 |

| 63 | .7524000000 | -.5577822765 | +.7771766386 |

| 63 | .7920000000 | -.5524681503 | +.8102866204 |

| 63 | .8316000000 | -.5419344965 | +.8345791246 |

| 63 | .8712000000 | -.5251401980 | +.8472261861 |

| 63 | .9108000000 | -.4995701088 | +.8426082503 |

| 63 | .9504000000 | -.4582764271 | +.8065665118 |

| 63 | .9900000000 | -.3631611966 | +.6657955272 |



2 : 1-Resonance in +.1343870049......

2 : 2-Resonance in +.2651460995......

2 : 3-Resonance in +.3888961596......

2 : 4-Resonance in +.5027329013......




--> 2 : 1 Resonance in .134387

--> kappa in [.084387, .184387]



-------------------------------------------------------------------------------------------


| 63 | .0843870049 | +0.0000000000 | -.3067816000 | certain |

| 63 | .1343870049 | +0.0000000000 | -.1167114676 | certain |

| 63 | .1843870049 | +0.0000000000 | +.2546683406 | certain |





--> 2 : 2 Resonance in .265146

--> kappa in [.215146, .315146]



-------------------------------------------------------------------------------------------


| 63 | .2151460995 | -.0318656728 | +.4544844035 | certain |

| 63 | .2651460995 | -.0348682456 | -.0000191174 | mostly |

| 63 | .3151460995 | -.0384116075 | -.2106683329 | certain |




--> 2 : 3 Resonance in .388896

--> kappa in [.338896, .438896]



-------------------------------------------------------------------------------------------


| 63 | .3388961596 | +0.0000000000 | -.2561885854 | certain |

| 63 | .3888961596 | +0.0000000000 | -.3306452525 | certain |

| 63 | .4388961596 | +0.0000000000 | -.3955309217 | certain |




--> 2 : 4 Resonance in .502732

--> kappa in [.452732, .552732]



-------------------------------------------------------------------------------------------


| 63 | .4527329013 | -.0017777368 | -.4119719930 | certain |

| 63 | .5027329013 | -.0023731029 | -.4642610553 | certain |

| 63 | .5527329013 | -.0031129066 | -.5041134724 | certain |


The calculations carried out for the upper and central domain yield

P

P=-1

Q

Figure 4.25: a = 0.54, = 0,m = 1

the same qualitative interpretation as in section 4.5.2. By ways ofcontrast the drift in the lower domain has two zeroes. The first atP ≈ −3.8 indicates a repulsive set while the second at P ≈ −2 cor-responds to an attractive area. As this zero and the 2 : 2 resonanceare close and possibly identical, the results of section 2.3 might notbe applicable here. However we note that P < −1 physically corre-sponds to negative values for d

dτ ϑ. Thus the solutions in the lowerdomain are attracted by a region implying backward rotation. As weconsider the case of an external torque m = 1 this is not surprising ifthe load is sufficiently large to compete with the force of the motor.


4.5.4 a = 4.1, = 0, m = 0

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0.2 0.4 0.6 0.8 1

G_0,0(kappa) ( abar=4.1 , rho=0 , m=0 , M=63)

-0.24

-0.22

-0.2

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0.2 0.4 0.6 0.8 1


0

0.1

0.2

0.3

0.4

0.5

0.2 0.4 0.6 0.8 1




G0,0(κ)−a/κ on

the lower domains


upper domain :


--> parameter a=4.1 rho=0 m=0

--> kappa in [0, .99]



--------------------------------------------------------------------------------


| 63 | .0396000000 | -.0095669817 | -.0000924030 |

| 63 | .0792000000 | -.0189572528 | -.0003661986 |

| 63 | .1188000000 | -.0281792944 | -.0008165122 |

| 63 | .1584000000 | -.0372402044 | -.0014387435 |

| 63 | .1980000000 | -.0461457952 | -.0022285042 |

| 63 | .2376000000 | -.0549006459 | -.0031815593 |

| 63 | .2772000000 | -.0635081130 | -.0042937680 |

| 63 | .3168000000 | -.0719702944 | -.0055610217 |

| 63 | .3564000000 | -.0802879409 | -.0069791761 |

| 63 | .3960000000 | -.0884603026 | -.0085439706 |

| 63 | .4356000000 | -.0964848886 | -.0102509310 |

| 63 | .4752000000 | -.1043571107 | -.0120952436 |

| 63 | .5148000000 | -.1120697574 | -.0140715880 |

| 63 | .5544000000 | -.1196122176 | -.0161739057 |

| 63 | .5940000000 | -.1269693146 | -.0183950665 |

| 63 | .6336000000 | -.1341195160 | -.0207263720 |

| 63 | .6732000000 | -.1410320897 | -.0231567811 |

| 63 | .7128000000 | -.1476623949 | -.0256716475 |

| 63 | .7524000000 | -.1539436566 | -.0282505383 |

| 63 | .7920000000 | -.1597715731 | -.0308631916 |

| 63 | .8316000000 | -.1649727806 | -.0334613083 |

| 63 | .8712000000 | -.1692316049 | -.0359596522 |

| 63 | .9108000000 | -.1718847586 | -.0381835702 |

| 63 | .9504000000 | -.1711115173 | -.0396644844 |

| 63 | .9900000000 | -.1552516904 | -.0374876032 |



2 : 1-Resonance in +.8038707404......

2 : 2-Resonance in +.9878039867......




--> 2 : 1 Resonance in .803870

--> kappa in [.753870, .853870]



-----------------------------------------------------------------------------------------


| 63 | .7538707404 | +0.0000000000 | -.1541690185 | certain |

| 63 | .8038707404 | +0.0000000000 | -.1614070977 | certain |

| 63 | .8538707404 | +0.0000000000 | -.1675141054 | certain |



*********************** Calculation of 2*|G_-k,2(kappa)| ***************************

--> parameter abar=4.1 rho=0 m=0

--> 2 : 2 Resonance in .987803

--> kappa in [.937803, .987803]



-----------------------------------------------------------------------------------------


| 63 | .9378039867 | +1.1010778373 | -.1719643559 | mostly |

| 63 | .9628039867 | +.5373629107 | -.1692869165 | mostly |

| 63 | .9878039867 | +.3885728462 | -.1576711399 | mostly |


We conclude that up to an O(ε)–set all orbits pass through the 2 : 2 resonance at κkm ≈ 0.988 situatedclose to the upper separatrix.

central domain :



--> kappa in [0, .99]



--------------------------------------------------------------------------------


| 63 | .0396000000 | -.0183348688 | -.1129272532 |

| 63 | .0792000000 | -.0364810169 | -.1123460733 |

| 63 | .1188000000 | -.0542535008 | -.1113851950 |

| 63 | .1584000000 | -.0714749993 | -.1100563551 |

| 63 | .1980000000 | -.0879797762 | -.1083761718 |

| 63 | .2376000000 | -.1036177765 | -.1063662812 |

| 63 | .2772000000 | -.1182588207 | -.1040534445 |

| 63 | .3168000000 | -.1317968033 | -.1014695763 |

| 63 | .3564000000 | -.1441537322 | -.0986516467 |

| 63 | .3960000000 | -.1552833893 | -.0956414075 |

| 63 | .4356000000 | -.1651743492 | -.0924849096 |

| 63 | .4752000000 | -.1738521029 | -.0892318012 |

| 63 | .5148000000 | -.1813801006 | -.0859344384 |

| 63 | .5544000000 | -.1878597129 | -.0826469014 |

| 63 | .5940000000 | -.1934294449 | -.0794240966 |

| 63 | .6336000000 | -.1982643239 | -.0763212629 |

| 63 | .6732000000 | -.2025773916 | -.0733944146 |

| 63 | .7128000000 | -.2066270581 | -.0707026423 |

| 63 | .7524000000 | -.2107375600 | -.0683139352 |

| 63 | .7920000000 | -.2153464056 | -.0663175676 |

| 63 | .8316000000 | -.2211019890 | -.0648476604 |

| 63 | .8712000000 | -.2290058009 | -.0641128023 |

| 63 | .9108000000 | -.2399676172 | -.0642607456 |

| 63 | .9504000000 | -.2444025627 | -.0627213606 |

| 63 | .9900000000 | -.1776456195 | -.0437658584 |



2 : 1-Resonance in +.3075848728......

2 : 2-Resonance in +.9867635460......





--> 2 : 1 Resonance in .307584

--> kappa in [.257584, .357584]



-----------------------------------------------------------------------------------------


| 63 | .2575848728 | +.1614991274 | -.1111385236 | mostly |

| 63 | .3075848728 | +.2323316952 | -.1287495137 | mostly |

| 63 | .3575848728 | +.3173595687 | -.1445046992 | mostly |


******************* Calculation of 2*|G_-k,2(kappa)| ***********************


--> 2 : 2 Resonance in .986763

--> kappa in [.936763, .986763]



-----------------------------------------------------------------------------------------


| 63 | .9367635460 | +.1340754295 | -.2461359773 | certain |

| 63 | .9617635460 | +.1540029415 | -.2354069917 | certain |

| 63 | .9867635460 | +.1516329740 | -.1838326901 | certain |


For this choice of parameters all solutions up to a set of size O(ε) pass through the 2 : 1 resonance atκkm ≈ 0.3 (i.e. P(hm) ≈ 1.23 of the central domain). The remaining solutions may possibly be capturedin this resonance. Recall that in the simulations described in section 4.2.2 we have observed such capturesfor a set which becomes smaller as ε→ 0 indeed.

lower domain :



--> kappa in [0, .99]



----------------------------------------------------------------------------------


| 63 | .0396000000 | +.0097536993 | -.0000942064 |

| 63 | .0792000000 | +.0197058472 | -.0003806592 |

| 63 | .1188000000 | +.0298701616 | -.0008655061 |

| 63 | .1584000000 | +.0402626922 | -.0015555147 |

| 63 | .1980000000 | +.0509022709 | -.0024582072 |

| 63 | .2376000000 | +.0618111018 | -.0035820287 |

| 63 | .2772000000 | +.0730155393 | -.0049365628 |

| 63 | .3168000000 | +.0845471273 | -.0065328121 |

| 63 | .3564000000 | +.0964440038 | -.0083835714 |

| 63 | .3960000000 | +.1087528323 | -.0105039320 |

| 63 | .4356000000 | +.1215315090 | -.0129119817 |

| 63 | .4752000000 | +.1348530409 | -.0156297963 |

| 63 | .5148000000 | +.1488112416 | -.0186848846 |

| 63 | .5544000000 | +.1635293431 | -.0221123580 |

| 63 | .5940000000 | +.1791734509 | -.0259582999 |

| 63 | .6336000000 | +.1959743907 | -.0302852131 |

| 63 | .6732000000 | +.2142648014 | -.0351812352 |

| 63 | .7128000000 | +.2345455049 | -.0407765941 |

| 63 | .7524000000 | +.2576118234 | -.0472749112 |

| 63 | .7920000000 | +.2848117979 | -.0550173034 |

| 63 | .8316000000 | +.3186159197 | -.0646246338 |

| 63 | .8712000000 | +.3639315869 | -.0773310240 |


| 63 | .9108000000 | +.4302297301 | -.0955739605 |

| 63 | .9504000000 | +.5033776069 | -.1166853847 |

| 63 | .9900000000 | +.1969846936 | -.0475645967 |



2 : 1-Resonance in +.8038707404......

2 : 2-Resonance in +.9878039867......




--> 2 : 1 Resonance in .803870

--> kappa in [.753870, .853870]



-------------------------------------------------------------------------------------------


| 63 | .7538707404 | +0.0000000000 | +.2585372574 | certain |

| 63 | .8038707404 | +0.0000000000 | +.2940933405 | certain |

| 63 | .8538707404 | +0.0000000000 | +.3422027229 | certain |




--> 2 : 2 Resonance in .987803

--> kappa in [.937803, .987803]



-------------------------------------------------------------------------------------------


| 63 | .9378039867 | -1.3260918711 | +.4876879515 | mostly |

| 63 | .9628039867 | -.7919794060 | +.4798487513 | mostly |

| 63 | .9878039867 | -.1268307864 | +.2199931224 | certain |


In the lower domain, all orbits pass the resonances arising and approach theP

P=-1

Q

Figure 4.27: a = 4.1, = 0, m = 0

lower separatrix region.


4.5.5 a = 4.1, = 1, m = 0

-100

-80

-60

-40

-20

00.2 0.4 0.6 0.8 1

G_0,0(kappa) ( abar=4.1 , rho=1.0 , m=0 , M=63)

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0.2 0.4 0.6 0.8 1


0

20

40

60

80

100

0.2 0.4 0.6 0.8 1




G0,0(κ)−a/κ on

the lower domains


upper domain :



--> kappa in [0, .99]



--------------------------------------------------------------------------------


| 63 | .0396000000 | -103.9635284755 | -1.0041355433 |

| 63 | .0792000000 | -52.1233606941 | -1.0068707724 |

| 63 | .1188000000 | -34.7942190763 | -1.0081837137 |

| 63 | .1584000000 | -26.0921659205 | -1.0080485565 |

| 63 | .1980000000 | -20.8403255244 | -1.0064352326 |

| 63 | .2376000000 | -17.3129898053 | -1.0033088726 |

| 63 | .2772000000 | -14.7704882524 | -.9986291081 |

| 63 | .3168000000 | -12.8429028620 | -.9923491772 |

| 63 | .3564000000 | -11.3246368136 | -.9844147708 |

| 63 | .3960000000 | -10.0922383712 | -.9747625353 |

| 63 | .4356000000 | -9.0670436852 | -.9633181047 |

| 63 | .4752000000 | -8.1964925365 | -.9499934764 |

| 63 | .5148000000 | -7.4440601230 | -.9346834515 |

| 63 | .5544000000 | -6.7834936337 | -.9172607001 |

| 63 | .5940000000 | -6.1953398511 | -.8975687491 |

| 63 | .6336000000 | -5.6647538438 | -.8754117159 |

| 63 | .6732000000 | -5.1800487132 | -.8505387301 |

| 63 | .7128000000 | -4.7316763584 | -.8226192459 |

| 63 | .7524000000 | -4.3114396356 | -.7912017516 |

| 63 | .7920000000 | -3.9117717760 | -.7556398162 |

| 63 | .8316000000 | -3.5248715434 | -.7149471159 |

| 63 | .8712000000 | -3.1412408050 | -.6674753632 |

| 63 | .9108000000 | -2.7461676113 | -.6100510878 |

| 63 | .9504000000 | -2.3069822442 | -.5347697377 |

| 63 | .9900000000 | -1.6581837158 | -.4003907021 |



2 : 1-Resonance in +.8038707404......

2 : 2-Resonance in +.9878039867......




--> 2 : 1 Resonance in .803870

--> kappa in [.753870, .853870]



-----------------------------------------------------------------------------------------

| M | kappa | G_-1,2(kappa)/(a/kappa)| | G_0,0(kappa)/(a/kappa)|| passage |

| 63 | .7538707404 | +0.0000000000 | -4.2962744771 | certain |

| 63 | .8038707404 | +0.0000000000 | -3.7948329322 | certain |

| 63 | .8538707404 | +0.0000000000 | -3.3094704714 | certain |





--> 2 : 2 Resonance in .987803

--> kappa in [.937803, .987803]



-----------------------------------------------------------------------------------------


| 63 | .9378039867 | +1.1010778373 | -2.4553353062 | certain |

| 63 | .9628039867 | +.5373629107 | -2.1465898444 | certain |

| 63 | .9878039867 | +.3885728462 | -1.7145290398 | certain |


central domain :



--> kappa in [0, .99]



--------------------------------------------------------------------------------


| 63 | .0396000000 | -0.0994989434 | -.6128291663 |

| 63 | .0792000000 | -0.1987133128 | -.6119527987 |

| 63 | .1188000000 | -0.2973607928 | -.6104968235 |

| 63 | .1584000000 | -0.3951635878 | -.6084682001 |

| 63 | .1980000000 | -0.4918506650 | -.6058766506 |

| 63 | .2376000000 | -0.5871599104 | -.6027345718 |

| 63 | .2772000000 | -0.6808400647 | -.5990568267 |

| 63 | .3168000000 | -0.7726522062 | -.5948603460 |

| 63 | .3564000000 | -0.8623704436 | -.5901634526 |

| 63 | .3960000000 | -0.9497813437 | -.5849848138 |

| 63 | .4356000000 | -1.0346814769 | -.5793419096 |

| 63 | .4752000000 | -1.1168722919 | -.5732488974 |

| 63 | .5148000000 | -1.1961513356 | -.5667137300 |

| 63 | .5544000000 | -1.2722985562 | -.5597343453 |

| 63 | .5940000000 | -1.3450559918 | -.5522936650 |

| 63 | .6336000000 | -1.4140983398 | -.5443529578 |

| 63 | .6732000000 | -1.4789903074 | -.5358427559 |

| 63 | .7128000000 | -1.5391232812 | -.5266497225 |

| 63 | .7524000000 | -1.5936162517 | -.5165960801 |

| 63 | .7920000000 | -1.6411466798 | -.5054036338 |

| 63 | .8316000000 | -1.6796184458 | -.4926202928 |

| 63 | .8712000000 | -1.7053376252 | -.4774288408 |

| 63 | .9108000000 | -1.7099435467 | -.4579044813 |

| 63 | .9504000000 | -1.6601301063 | -.4260414373 |

| 63 | .9900000000 | -1.3641773148 | -.3360870448 |



2 : 1-Resonance in +.3075848728......

2 : 2-Resonance in +.9867635460......





--> 2 : 1 Resonance in .307584

--> kappa in [.257584, .357584]



-----------------------------------------------------------------------------------------


| 63 | .2575848728 | +.1614991274 | -.6346562326 | certain |

| 63 | .3075848728 | +.2323316952 | -.7514659023 | certain |

| 63 | .3575848728 | +.3173595687 | -.8650203885 | certain |




--> 2 : 2 Resonance in .986763

--> kappa in [.936763, .986763]



-----------------------------------------------------------------------------------------


| 63 | .9367635460 | +.1340754295 | -1.6889205617 | certain |

| 63 | .9617635460 | +.1540029415 | -1.6175647928 | certain |

| 63 | .9867635460 | +.1516329740 | -1.4111461809 | certain |


lower domain :



--> kappa in [0, .99]



----------------------------------------------------------------------------------


| 63 | .0396000000 | +102.9641074254 | -.9944825985 |

| 63 | .0792000000 | +51.1256805330 | -.9875985117 |

| 63 | .1188000000 | +33.7994539872 | -.9793597887 |

| 63 | .1584000000 | +25.1015109210 | -.9697754463 |

| 63 | .1980000000 | +19.8550057333 | -.9588514963 |

| 63 | .2376000000 | +16.3342707219 | -.9465909081 |

| 63 | .2772000000 | +13.7996874693 | -.9329935040 |

| 63 | .3168000000 | +11.8814037494 | -.9180557823 |

| 63 | .3564000000 | +10.3739048950 | -.9017706596 |

| 63 | .3960000000 | +9.1538413796 | -.8841271186 |

| 63 | .4356000000 | +8.1426767014 | -.8651097490 |

| 63 | .4752000000 | +7.2880102489 | -.8446981634 |

| 63 | .5148000000 | +6.5535192746 | -.8228662738 |

| 63 | .5544000000 | +5.9132104045 | -.7995814264 |

| 63 | .5940000000 | +5.3479698130 | -.7748034314 |

| 63 | .6336000000 | +4.8434072866 | -.7484836236 |

| 63 | .6732000000 | +4.3884638371 | -.7205643549 |

| 63 | .7128000000 | +3.9744921582 | -.6909800025 |

| 63 | .7524000000 | +3.5946514947 | -.6596623864 |

| 63 | .7920000000 | +3.2435474970 | -.6265584433 |

| 63 | .8316000000 | +2.9171399311 | -.5916813577 |

| 63 | .8712000000 | +2.6130509022 | -.5552414502 |

| 63 | .9108000000 | +2.3304129569 | -.5176927124 |

| 63 | .9504000000 | +2.0336280953 | -.4714049126 |

| 63 | .9900000000 | +1.2319443416 | -.2974694873 |




2 : 1-Resonance in +.8038707404......

2 : 2-Resonance in +.9878039867......




--> 2 : 1 Resonance in .803870

--> kappa in [.753870, .853870]



-------------------------------------------------------------------------------------------


| 63 | .7538707404 | +0.0000000000 | +3.5811268746 | certain |

| 63 | .8038707404 | +0.0000000000 | +3.1432550380 | certain |

| 63 | .8538707404 | +0.0000000000 | +2.7434415410 | certain |




--> 2 : 2 Resonance in .987803

--> kappa in [.937803, .987803]



-------------------------------------------------------------------------------------------


| 63 | .9378039867 | -1.3260918711 | +2.1403696584 | certain |

| 63 | .9628039867 | -.7919794060 | +1.8811404240 | certain |

| 63 | .9878039867 | -.1268307864 | +1.2949954190 | certain |


Since is sufficiently large all solutions pass through the resonances appear-P

P=-1

Q

Figure 4.29: a = 4.1, = 1, m = 0

ing in each of the three domains. The solutions of the upper and lowerdomains tend towards the separatrices while the periodic solution near theorigin is globally attractive on the central domain.


4.5.6 a = 4.1, = 0, m = 1

-0.58

-0.56

-0.54

-0.52

-0.5

-0.48

-0.46

-0.44

-0.42

-0.4

0.2 0.4 0.6 0.8 1

G_0,0(kappa) ( abar=4.1 , rho=0 , m=1.0 , M=63)

-0.24

-0.22

-0.2

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0.2 0.4 0.6 0.8 1

G_0,0(kappa) ( abar=4.1 , rho=0 , m=1. , M=63)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.2 0.4 0.6 0.8 1




G0,0(κ)−a/κ on

the lower domains


upper domain :



--> kappa in [0, .99]



--------------------------------------------------------------------------------


| 63 | .0396000000 | -.5093708655 | -.0049197771 |

| 63 | .0792000000 | -.5181716306 | -.0100095593 |

| 63 | .1188000000 | -.5264072726 | -.0152529717 |

| 63 | .1584000000 | -.5340789481 | -.0206336842 |

| 63 | .1980000000 | -.5411839286 | -.0261352238 |

| 63 | .2376000000 | -.5477154156 | -.0317407762 |

| 63 | .2772000000 | -.5536622177 | -.0374329675 |

| 63 | .3168000000 | -.5590082671 | -.0431936144 |

| 63 | .3564000000 | -.5637319317 | -.0490034293 |

| 63 | .3960000000 | -.5678050632 | -.0548416597 |

| 63 | .4356000000 | -.5711916907 | -.0606856342 |

| 63 | .4752000000 | -.5738462196 | -.0665101764 |

| 63 | .5148000000 | -.5757109237 | -.0722868252 |

| 63 | .5544000000 | -.5767123950 | -.0779827687 |

| 63 | .5940000000 | -.5767564018 | -.0835593421 |

| 63 | .6336000000 | -.5757202320 | -.0889698387 |

| 63 | .6732000000 | -.5734408836 | -.0941561958 |

| 63 | .7128000000 | -.5696960500 | -.0990437425 |

| 63 | .7524000000 | -.5641718105 | -.1035324073 |

| 63 | .7920000000 | -.5564038250 | -.1074809340 |

| 63 | .8316000000 | -.5456601563 | -.1106758502 |

| 63 | .8712000000 | -.5306765473 | -.1127622946 |

| 63 | .9108000000 | -.5089345715 | -.1130579531 |

| 63 | .9504000000 | -.4739216366 | -.1098573471 |

| 63 | .9900000000 | -.3892378791 | -.0939867073 |



2 : 1-Resonance in +.8038707404......

2 : 2-Resonance in +.9878039867......




--> 2 : 1 Resonance in .803870

--> kappa in [.753870, .853870]



-----------------------------------------------------------------------------------------


| 63 | .7538707404 | +0.0000000000 | -.5639269391 | certain |

| 63 | .8038707404 | +0.0000000000 | -.5535391662 | certain |

| 63 | .8538707404 | +0.0000000000 | -.5378728793 | certain |





--> 2 : 2 Resonance in .987803

--> kappa in [.937803, .987803]



-----------------------------------------------------------------------------------------


| 63 | .9378039867 | +1.1010778373 | -.4873089776 | mostly |

| 63 | .9628039867 | +.5373629107 | -.4572925441 | mostly |

| 63 | .9878039867 | +.3885728462 | -.3985989415 | certain |


central domain :



--> kappa in [0, .99]



--------------------------------------------------------------------------------


| 63 | .0396000000 | -.0183348688 | -.1129272532 |

| 63 | .0792000000 | -.0364810169 | -.1123460733 |

| 63 | .1188000000 | -.0542535008 | -.1113851950 |

| 63 | .1584000000 | -.0714749993 | -.1100563551 |

| 63 | .1980000000 | -.0879797762 | -.1083761718 |

| 63 | .2376000000 | -.1036177765 | -.1063662812 |

| 63 | .2772000000 | -.1182588207 | -.1040534445 |

| 63 | .3168000000 | -.1317968033 | -.1014695763 |

| 63 | .3564000000 | -.1441537322 | -.0986516467 |

| 63 | .3960000000 | -.1552833893 | -.0956414075 |

| 63 | .4356000000 | -.1651743492 | -.0924849096 |

| 63 | .4752000000 | -.1738521029 | -.0892318012 |

| 63 | .5148000000 | -.1813801006 | -.0859344384 |

| 63 | .5544000000 | -.1878597129 | -.0826469014 |

| 63 | .5940000000 | -.1934294449 | -.0794240966 |

| 63 | .6336000000 | -.1982643239 | -.0763212629 |

| 63 | .6732000000 | -.2025773916 | -.0733944146 |

| 63 | .7128000000 | -.2066270581 | -.0707026423 |

| 63 | .7524000000 | -.2107375600 | -.0683139352 |

| 63 | .7920000000 | -.2153464056 | -.0663175676 |

| 63 | .8316000000 | -.2211019890 | -.0648476604 |

| 63 | .8712000000 | -.2290058009 | -.0641128023 |

| 63 | .9108000000 | -.2399676172 | -.0642607456 |

| 63 | .9504000000 | -.2444025627 | -.0627213606 |

| 63 | .9900000000 | -.1776456195 | -.0437658584 |



2 : 1-Resonance in +.3075848728......

2 : 2-Resonance in +.9867635460......





--> 2 : 1 Resonance in .307584

--> kappa in [.257584, .357584]



-----------------------------------------------------------------------------------------


| 63 | .2575848728 | +.1614991274 | -.1111385236 | mostly |

| 63 | .3075848728 | +.2323316952 | -.1287495137 | mostly |

| 63 | .3575848728 | +.3173595687 | -.1445046992 | mostly |




--> 2 : 2 Resonance in .986763

--> kappa in [.936763, .986763]



-----------------------------------------------------------------------------------------


| 63 | .9367635460 | +.1340754295 | -.2461359773 | certain |

| 63 | .9617635460 | +.1540029415 | -.2354069917 | certain |

| 63 | .9867635460 | +.1516329740 | -.1838326901 | certain |


lower domain :



--> kappa in [0, .99]



----------------------------------------------------------------------------------


| 63 | .0396000000 | -.4900501845 | +.0047331676 |

| 63 | .0792000000 | -.4795085305 | +.0092627013 |

| 63 | .1188000000 | -.4683578165 | +.0135709533 |

| 63 | .1584000000 | -.4565760513 | +.0176394259 |

| 63 | .1980000000 | -.4441358623 | +.0214485123 |

| 63 | .2376000000 | -.4310036677 | +.0249771881 |

| 63 | .2772000000 | -.4171385653 | +.0282026366 |

| 63 | .3168000000 | -.4024908454 | +.0310997804 |

| 63 | .3564000000 | -.3869999868 | +.0336406817 |

| 63 | .3960000000 | -.3705919282 | +.0357937569 |

| 63 | .4356000000 | -.3531752930 | +.0375227213 |

| 63 | .4752000000 | -.3346360680 | +.0387851364 |

| 63 | .5148000000 | -.3148299246 | +.0395303524 |

| 63 | .5544000000 | -.2935708342 | +.0396965049 |

| 63 | .5940000000 | -.2706136362 | +.0392059755 |

| 63 | .6336000000 | -.2456263252 | +.0379582535 |

| 63 | .6732000000 | -.2181439924 | +.0358181794 |

| 63 | .7128000000 | -.1874881501 | +.0325955008 |

| 63 | .7524000000 | -.1526163304 | +.0280069578 |

| 63 | .7920000000 | -.1118204539 | +.0216004389 |

| 63 | .8316000000 | -.0620714559 | +.0125899079 |

| 63 | .8712000000 | +.0024866445 | -.0005283816 |

| 63 | .9108000000 | +.0931799172 | -.0206995777 |

| 63 | .9504000000 | +.2005674876 | -.0464925220 |

| 63 | .9900000000 | -.0370014950 | +.0089345073 |




2 : 1-Resonance in +.8038707404......

2 : 2-Resonance in +.9878039867......




--> 2 : 1 Resonance in .803870

--> kappa in [.753870, .853870]



-------------------------------------------------------------------------------------------


| 63 | .7538707404 | +0.0000000000 | -.1512206632 | certain |

| 63 | .8038707404 | +0.0000000000 | -.0980387279 | certain |

| 63 | .8538707404 | +0.0000000000 | -.0281560509 | certain |




--> 2 : 2 Resonance in .987803

--> kappa in [.937803, .987803]



-------------------------------------------------------------------------------------------


| 63 | .9378039867 | -1.3260918711 | +.1723433298 | mostly |

| 63 | .9628039867 | -.7919794060 | +.1918431237 | mostly |

| 63 | .9878039867 | -.1268307864 | -.0209346791 | mostly |


The situation is qualitatively equivalent to the one observed in section 4.5.4P

P=-1

Q

Figure 4.31: a = 4.1, = 0, m = 1

for = 0, m = 0. A passage up to a O(ε) set, however, arises in the 2 : 2resonance at κkm ≈ 0.98 situated close to the lower separatrix here.


4.5.7 a = 20.38, = 0, m = 0

-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0.2 0.4 0.6 0.8 1


-0.05

-0.04

-0.03

-0.02

-0.01

0.2 0.4 0.6 0.8 1


0.01

0.02

0.03

0.04

0.2 0.4 0.6 0.8 1




G0,0(κ)−a/κ



upper domain :



--> kappa in [0, .99]



--------------------------------------------------------------------------------


| 63 | .0396000000 | -.0019393189 | -.0000037682 |

| 63 | .0792000000 | -.0038711513 | -.0000150439 |

| 63 | .1188000000 | -.0057955195 | -.0000337834 |

| 63 | .1584000000 | -.0077123913 | -.0000599432 |

| 63 | .1980000000 | -.0096216679 | -.0000934784 |

| 63 | .2376000000 | -.0115231695 | -.0001343427 |

| 63 | .2772000000 | -.0134166167 | -.0001824870 |

| 63 | .3168000000 | -.0153016055 | -.0002378581 |

| 63 | .3564000000 | -.0171775757 | -.0003003968 |

| 63 | .3960000000 | -.0190437667 | -.0003700358 |

| 63 | .4356000000 | -.0208991584 | -.0004466964 |

| 63 | .4752000000 | -.0227423881 | -.0005302837 |

| 63 | .5148000000 | -.0245716332 | -.0006206809 |

| 63 | .5544000000 | -.0263844401 | -.0007177396 |

| 63 | .5940000000 | -.0281774695 | -.0008212667 |

| 63 | .6336000000 | -.0299461040 | -.0009310035 |

| 63 | .6732000000 | -.0316838219 | -.0010465921 |

| 63 | .7128000000 | -.0333811511 | -.0011675213 |

| 63 | .7524000000 | -.0350238227 | -.0012930286 |

| 63 | .7920000000 | -.0365892724 | -.0014219187 |

| 63 | .8316000000 | -.0380393704 | -.0015521854 |

| 63 | .8712000000 | -.0393032314 | -.0016801263 |

| 63 | .9108000000 | -.0402279060 | -.0017978202 |

| 63 | .9504000000 | -.0403803780 | -.0018830967 |

| 63 | .9900000000 | -.0370744636 | -.0018009675 |




central domain :



--> kappa in [0, .99]



--------------------------------------------------------------------------------

| M | kappa | G_0,0(kappa)/(a*kappa) | G_0,0(kappa)/(kappa*a)^2 |

| 63 | .0396000000 | -.0039625525 | -.0049099341 |

| 63 | .0792000000 | -.0079113463 | -.0049014100 |

| 63 | .1188000000 | -.0118325499 | -.0048871731 |

| 63 | .1584000000 | -.0157121835 | -.0048671775 |

| 63 | .1980000000 | -.0195360393 | -.0048413574 |

| 63 | .2376000000 | -.0232895935 | -.0048096258 |

| 63 | .2772000000 | -.0269579065 | -.0047718717 |

| 63 | .3168000000 | -.0305255085 | -.0047279573 |

| 63 | .3564000000 | -.0339762616 | -.0046777145 |

| 63 | .3960000000 | -.0372931942 | -.0046209388 |

| 63 | .4356000000 | -.0404582925 | -.0045573827 |

| 63 | .4752000000 | -.0434522354 | -.0044867462 |

| 63 | .5148000000 | -.0462540484 | -.0044086643 |

| 63 | .5544000000 | -.0488406378 | -.0043226883 |

| 63 | .5940000000 | -.0511861489 | -.0042282614 |


| 63 | .6336000000 | -.0532610487 | -.0041246809 |

| 63 | .6732000000 | -.0550307682 | -.0040110427 |

| 63 | .7128000000 | -.0564535952 | -.0038861515 |

| 63 | .7524000000 | -.0574772234 | -.0037483731 |

| 63 | .7920000000 | -.0580326839 | -.0035953675 |

| 63 | .8316000000 | -.0580226497 | -.0034235675 |

| 63 | .8712000000 | -.0572958240 | -.0032270145 |

| 63 | .9108000000 | -.0555790142 | -.0029942195 |

| 63 | .9504000000 | -.0522260064 | -.0026963496 |

| 63 | .9900000000 | -.0439543128 | -.0021785228 |




lower domain :



--> kappa in [0, .99]



----------------------------------------------------------------------------------


| 63 | .0396000000 | +.0019468731 | -.0000037829 |

| 63 | .0792000000 | +.0039014051 | -.0000151614 |

| 63 | .1188000000 | +.0058637304 | -.0000341811 |

| 63 | .1584000000 | +.0078340060 | -.0000608884 |

| 63 | .1980000000 | +.0098124046 | -.0000953315 |

| 63 | .2376000000 | +.0117991063 | -.0001375597 |

| 63 | .2772000000 | +.0137942889 | -.0001876239 |

| 63 | .3168000000 | +.0157981158 | -.0002455762 |

| 63 | .3564000000 | +.0178107192 | -.0003114691 |

| 63 | .3960000000 | +.0198321780 | -.0003853553 |

| 63 | .4356000000 | +.0218624855 | -.0004672864 |

| 63 | .4752000000 | +.0239015033 | -.0005573108 |

| 63 | .5148000000 | +.0259488925 | -.0006554705 |

| 63 | .5544000000 | +.0280040098 | -.0007617970 |

| 63 | .5940000000 | +.0300657458 | -.0008763028 |

| 63 | .6336000000 | +.0321322654 | -.0009989697 |

| 63 | .6732000000 | +.0342005769 | -.0011297266 |

| 63 | .7128000000 | +.0362657837 | -.0012684126 |

| 63 | .7524000000 | +.0383197137 | -.0014147081 |

| 63 | .7920000000 | +.0403482266 | -.0015679978 |

| 63 | .8316000000 | +.0423254104 | -.0017270761 |

| 63 | .8712000000 | +.0441993543 | -.0018894248 |

| 63 | .9108000000 | +.0458497570 | -.0020490656 |

| 63 | .9504000000 | +.0469103056 | -.0021876130 |

| 63 | .9900000000 | +.0448817581 | -.0021802227 |





4.5.8 a = 20.38, = 1, m = 0

-500

-400

-300

-200

-100

00.2 0.4 0.6 0.8 1


-7

-6

-5

-4

-3

-2

-1

0.2 0.4 0.6 0.8 1


0

100

200

300

400

500

0.2 0.4 0.6 0.8 1




G0,0(κ)−a/κ



upper domain :



--> kappa in [0, .99]



--------------------------------------------------------------------------------


| 63 | .0396000000 | -514.7446046884 | -1.0001906940 |

| 63 | .0792000000 | -257.0186350694 | -.9988162854 |

| 63 | .1188000000 | -170.8401221711 | -.9958688181 |

| 63 | .1584000000 | -127.5469444271 | -.9913364081 |

| 63 | .1980000000 | -101.4062465628 | -.9852029842 |

| 63 | .2376000000 | -83.8400189303 | -.9774479145 |

| 63 | .2772000000 | -71.1715986696 | -.9680454931 |

| 63 | .3168000000 | -61.5622839303 | -.9569642565 |

| 63 | .3564000000 | -53.9901926624 | -.9441660777 |

| 63 | .3960000000 | -47.8417911655 | -.9296049706 |

| 63 | .4356000000 | -42.7262068503 | -.9132255006 |

| 63 | .4752000000 | -38.3823612383 | -.8949606506 |

| 63 | .5148000000 | -34.6289339413 | -.8747289103 |

| 63 | .5544000000 | -31.3357289214 | -.8524302313 |

| 63 | .5940000000 | -28.4064354584 | -.8279402680 |

| 63 | .6336000000 | -25.7677675539 | -.8011019392 |

| 63 | .6732000000 | -23.3623042177 | -.7717126201 |

| 63 | .7128000000 | -21.1435022902 | -.7395038485 |

| 63 | .7524000000 | -19.0719149710 | -.7041074005 |

| 63 | .7920000000 | -17.1118553878 | -.6649945764 |

| 63 | .8316000000 | -15.2275872507 | -.6213572893 |

| 63 | .8712000000 | -13.3771627036 | -.5718441681 |

| 63 | .9108000000 | -11.4979629262 | -.5138540055 |

| 63 | .9504000000 | -9.4548429811 | -.4409167207 |

| 63 | .9900000000 | -6.5786499577 | -.3195713178 |




central domain :



--> kappa in [0, .99]



--------------------------------------------------------------------------------


| 63 | .0396000000 | -0.4074073917 | -.5048118472 |

| 63 | .0792000000 | -0.8143245635 | -.5045081355 |

| 63 | .1188000000 | -1.2202536745 | -.5039988016 |

| 63 | .1584000000 | -1.6246813141 | -.5032790224 |

| 63 | .1980000000 | -2.0270698716 | -.5023418363 |

| 63 | .2376000000 | -2.4268478103 | -.5011779163 |

| 63 | .2772000000 | -2.8233983338 | -.4997752539 |

| 63 | .3168000000 | -3.2160457796 | -.4981187270 |

| 63 | .3564000000 | -3.6040388413 | -.4961895205 |

| 63 | .3960000000 | -3.9865293677 | -.4939643450 |

| 63 | .4356000000 | -4.3625449417 | -.4914143826 |

| 63 | .4752000000 | -4.7309525895 | -.4885038425 |

| 63 | .5148000000 | -5.0904096020 | -.4851879558 |

| 63 | .5544000000 | -5.4392951812 | -.4814101322 |

| 63 | .5940000000 | -5.7756127408 | -.4770978298 |


| 63 | .6336000000 | -6.0968457423 | -.4721563759 |

| 63 | .6732000000 | -6.3997369203 | -.4664593840 |

| 63 | .7128000000 | -6.6799348211 | -.4598332318 |

| 63 | .7524000000 | -6.9313961838 | -.4520305179 |

| 63 | .7920000000 | -7.1453033150 | -.4426814337 |

| 63 | .8316000000 | -7.3079166473 | -.4311962000 |

| 63 | .8712000000 | -7.3957452335 | -.4165430530 |

| 63 | .9108000000 | -7.3624349761 | -.3966379552 |

| 63 | .9504000000 | -7.0894277966 | -.3660164263 |

| 63 | .9900000000 | -5.9418850324 | -.2944997091 |




lower domain :



--> kappa in [0, .99]



----------------------------------------------------------------------------------


| 63 | .0396000000 | +513.7450044749 | -.9982483894 |

| 63 | .0792000000 | +256.0202365678 | -.9949363462 |

| 63 | .1188000000 | +169.8437344256 | -.9900606305 |

| 63 | .1584000000 | +126.5533885546 | -.9836141681 |

| 63 | .1980000000 +100.4163610328 | -.9755858432 |

| 63 | .2376000000 | +82.8546653279 | -.9659601806 |

| 63 | .2772000000 | +70.1916681324 | -.9547168992 |

| 63 | .3168000000 | +60.5887044950 | -.9418303034 |

| 63 | .3564000000 | +53.0239378244 | -.9272684710 |

| 63 | .3960000000 | +46.8838900555 | -.9109921718 |

| 63 | .4356000000 | +41.7777565731 | -.8929534231 |

| 63 | .4752000000 | +37.4445421356 | -.8730935438 |

| 63 | .5148000000 | +33.7030288680 | -.8513404936 |

| 63 | .5544000000 | +30.4231481363 | -.8276051681 |

| 63 | .5940000000 | +27.5087495603 | -.8017761157 |

| 63 | .6336000000 | +24.8867522833 | -.7737117883 |

| 63 | .6732000000 | +22.5000033848 | -.7432287673 |

| 63 | .7128000000 | +20.3023196126 | -.7100830922 |

| 63 | .7524000000 | +18.2547545543 | -.6739390248 |

| 63 | .7920000000 | +16.3223498381 | -.6343131046 |

| 63 | .8316000000 | +14.4704985392 | -.5904645036 |

| 63 | .8712000000 | +12.6591689417 | -.5411515202 |

| 63 | .9108000000 | +10.8294851515 | -.4839791499 |

| 63 | .9504000000 | +8.8557526701 | -.4129787702 |

| 63 | .9900000000 | +6.1184848749 | -.2972178619 |





4.5.9 a = 20.38, = 0, m = 1

-0.5

-0.45

-0.4

-0.35

-0.3

0.2 0.4 0.6 0.8 1


-0.05

-0.04

-0.03

-0.02

-0.01

0.2 0.4 0.6 0.8 1


-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

0.2 0.4 0.6 0.8 1




G0,0(κ)−a/κ



upper domain :



--> kappa in [0, .99]



--------------------------------------------------------------------------------


| 63 | .0396000000 | -.5017432028 | -.0009749279 |

| 63 | .0792000000 | -.5030855290 | -.0019550723 |

| 63 | .1188000000 | -.5040234977 | -.0029380761 |

| 63 | .1584000000 | -.5045511349 | -.0039215358 |

| 63 | .1980000000 | -.5046598013 | -.0049029754 |

| 63 | .2376000000 | -.5043379391 | -.0058798181 |

| 63 | .2772000000 | -.5035707214 | -.0068493525 |

| 63 | .3168000000 | -.5023395783 | -.0078086937 |

| 63 | .3564000000 | -.5006215664 | -.0087547363 |

| 63 | .3960000000 | -.4983885273 | -.0096840950 |

| 63 | .4356000000 | -.4956059605 | -.0105930302 |

| 63 | .4752000000 | -.4922314970 | -.0114773507 |

| 63 | .5148000000 | -.4882127995 | -.0123322840 |

| 63 | .5544000000 | -.4834846175 | -.0131522999 |

| 63 | .5940000000 | -.4779645567 | -.0139308609 |

| 63 | .6336000000 | -.4715468200 | -.0146600620 |

| 63 | .6732000000 | -.4640926158 | -.0153300858 |

| 63 | .7128000000 | -.4554148062 | -.0159283451 |

| 63 | .7524000000 | -.4452519765 | -.0164380562 |

| 63 | .7920000000 | -.4332215243 | -.0168356941 |

| 63 | .8316000000 | -.4187267462 | -.0170860236 |

| 63 | .8712000000 | -.4007481738 | -.0171310995 |

| 63 | .9108000000 | -.3772777189 | -.0168608707 |

| 63 | .9504000000 | -.3431904973 | -.0160043301 |

| 63 | .9900000000 | -.2710606523 | -.0131673231 |




central domain :



--> kappa in [0, .99]



--------------------------------------------------------------------------------


| 63 | .0396000000 | -.0039625525 | -.0049099341 |

| 63 | .0792000000 | -.0079113463 | -.0049014100 |

| 63 | .1188000000 | -.0118325499 | -.0048871731 |

| 63 | .1584000000 | -.0157121835 | -.0048671775 |

| 63 | .1980000000 | -.0195360393 | -.0048413574 |

| 63 | .2376000000 | -.0232895935 | -.0048096258 |

| 63 | .2772000000 | -.0269579065 | -.0047718717 |

| 63 | .3168000000 | -.0305255085 | -.0047279573 |

| 63 | .3564000000 | -.0339762616 | -.0046777145 |

| 63 | .3960000000 | -.0372931942 | -.0046209388 |

| 63 | .4356000000 | -.0404582925 | -.0045573827 |

| 63 | .4752000000 | -.0434522354 | -.0044867462 |

| 63 | .5148000000 | -.0462540484 | -.0044086643 |

| 63 | .5544000000 | -.0488406378 | -.0043226883 |

| 63 | .5940000000 | -.0511861489 | -.0042282614 |


| 63 | .6336000000 | -.0532610487 | -.0041246809 |

| 63 | .6732000000 | -.0550307682 | -.0040110427 |

| 63 | .7128000000 | -.0564535952 | -.0038861515 |

| 63 | .7524000000 | -.0574772234 | -.0037483731 |

| 63 | .7920000000 | -.0580326839 | -.0035953675 |

| 63 | .8316000000 | -.0580226497 | -.0034235675 |

| 63 | .8712000000 | -.0572958240 | -.0032270145 |

| 63 | .9108000000 | -.0555790142 | -.0029942195 |

| 63 | .9504000000 | -.0522260064 | -.0026963496 |

| 63 | .9900000000 | -.0439543128 | -.0021785228 |




lower domain :



--> kappa in [0, .99]



----------------------------------------------------------------------------------


| 63 | .0396000000 | -.4978570106 | +.0009673767 |

| 63 | .0792000000 | -.4953129725 | +.0019248669 |

| 63 | .1188000000 | -.4923642477 | +.0028701115 |

| 63 | .1584000000 | -.4890047375 | +.0038007041 |

| 63 | .1980000000 | -.4852257287 | +.0047141655 |

| 63 | .2376000000 | -.4810156632 | +.0056079156 |

| 63 | .2772000000 | -.4763598157 | +.0064792414 |

| 63 | .3168000000 | -.4712398569 | +.0073252594 |

| 63 | .3564000000 | -.4656332715 | +.0081428703 |

| 63 | .3960000000 | -.4595125825 | +.0089287037 |

| 63 | .4356000000 | -.4528443165 | +.0096790473 |

| 63 | .4752000000 | -.4455876055 | +.0103897561 |

| 63 | .5148000000 | -.4376922737 | +.0110561326 |

| 63 | .5544000000 | -.4290961675 | +.0116727632 |

| 63 | .5940000000 | -.4197213413 | +.0122332913 |

| 63 | .6336000000 | -.4094684505 | +.0127300888 |

| 63 | .6732000000 | -.3982082170 | +.0131537670 |

| 63 | .7128000000 | -.3857678713 | +.0134924111 |

| 63 | .7524000000 | -.3719084400 | +.0137303194 |

| 63 | .7920000000 | -.3562840252 | +.0138457776 |

| 63 | .8316000000 | -.3383619652 | +.0138067620 |

| 63 | .8712000000 | -.3172455880 | +.0135615483 |

| 63 | .9108000000 | -.2912000558 | +.0130139848 |

| 63 | .9504000000 | -.2558998136 | +.0119336203 |

| 63 | .9900000000 | -.1891044304 | +.0091861327 |





The qualitative behaviour for a = 20.38 is simple : The solutions contained in theP

P=-1

Q

Figure 4.35:a = 20.38

upper and lower domain tend towards the bound of these domains, close to the sep-aratrices. The periodic solution near the origin is globally attractive in the centraldomain. There exist no resonances for κ ∈ [0, 0.99] as the resonances are extremlyclose to the separatrices (hence with values where κ is almost 1, cf. (4.58 c)).As the set P = −1 is contained in the central domain in the main, almost allsolutions with initial values equivalent to a startup of the synchronous motor (andtherefore P = −1) tend towards the origin (Q,P ) = (0, 0) or equivalently to the pe-riodic solution (q(t, ε), p(t, ε)). This, however, corresponds to a rotation of the motorwith constant frequency and small periodic perturbations in the angular speed d

dtϑ(cf. lemma 4.1.2). Even the addition of a linear damping ( 6= 0) or an external torque(m 6= 0) has no significant influence and does not change the qualitative behaviour.

4.6. The Stability of h = 0, Following Chapter 3 217

4.6 The Stability of h = 0, Following Chapter 3

In order to apply the results derived in corollary 3.2.3 let us determine the leading ε–terms of the quantityg,10,0(ε) :

Recalling (4.34) and (4.35) we see

F (Q,P, 0, t, ε) = ε F 1(Q,P, 0, t) + ε2 F 2(Q,P, 0, t) +O(ε3)

= ε2

[

−

(

0P

)

− (m+ )

(

01− cos(Q)

)

+[

z00 + z+0 eiQ + z−0 e

−iQ]

+[

z02 + z+2 eiQ + z−2 e

−iQ] ei2t +[

z0−2 + z+−2 eiQ + z−−2 e

−iQ] e−i2t]

+O(ε3)

such that

∂QF (0, 0, 0, t, ε) = ε2[

i[

z+0 − z−0]

+ i[

z+2 − z−2]

ei2t + i[

z+−2 − z−−2

]

e−i2t]

+O(ε3)

∂P F (0, 0, 0, t, ε) = −ε2 (

01

)

+O(ε3).

The definition (3.6) of g2,10,0 together with (1.162) then yields

g,10,0(ε) =1

(2π)2

2π∫

0

2π∫

0

g,1(t, ϕ, ε) dt dϕ =1

2π

2π∫

0

g,10 (t, ε) dt

=1

2π

∫ 2π

0

12

(

∂QFq(0, 0, 0, t, ε) + ∂P Fp(0, 0, 0, t, ε))

dt

= ε21

2π

∫ 2π

0

12

((

10

)∣

∣

∣

∣

i[

z+0 − z−0]

+ i[

z+2 − z−2]

ei2t + i[

z+−2 − z−−2

]

e−i2t)

− /2 dt+O(ε3)

= ε2

[

i

2

((

10

)∣

∣

∣

∣

[

z+0 − z−0]

)

− /2

]

+O(ε3).

Writing g,10,0(ε) = ε2 g2,10,0 +O(ε3) the corresponding evaluation with Maple [15] yields

g2,10,0 =−2 a10 + 64 a8 − 736 a6 − 32768− 14336 a2 + 512 a4

32768 a2 − 40 a10 + 528 a8 + 65536 + 30720 a4 + a12 − 2048 a6− /2.

As the value g2,10,0 is negative for all parameters a ∈ [0, 20.38] (cf. figure 4.36) and ≥ 0, the set h = 0(corresponding to the periodic solution of (4.14)) is linearly stable provided that |ε| is sufficiently smallto fulfill the theoretical considerations carried out in chapter 3.


-0.5

-0.4

-0.3

-0.2

-0.1

05 10 15 20

Figure 4.36: Graph of g2,10,0 for a ∈ [0, 20.38] and = 0

4.7. The Regions near the Separatrices 219

4.7 The Regions near the Separatrices

The aim of this last section is to discuss the regions close to the separatrix solutions of the unperturbedsystem of (4.14) (i.e. for ε = 0). As we will see in the first subsection the existence of a global, attractiveinvariant manifold near η = 0 may be established for (4.14) directly, provided that the parameter ais sufficiently small. Although this does not coincide with the parameter range under consideration,the investigation of the corresponding reduced system enables the explication of a phenomena found insection 4.2.2.

The process carried out here is not put into a general framework but demonstrated in the particularsituation of the miniature motor i.e. system (4.14). However the way followed illustrates the idea of proofand makes it possible to adapt or generalize it in similar situations.

4.7.1 The Existence of a Global, Attractive Invariant Manifold

Let us consider the equation (4.14) derived in section 4.1. Adding the equation ddst = 1 we write (4.14)

in autonomous form. Setting ξ = (t, q, p), y = η this autonomous representation is of the same form as(1.134) where

f0(ξ) =

1p

−(

a2

)2sin(q)

f1(ξ, y, ε) =

00

ε (η1 cos(q + t)− η2 sin(q + t))− ε2 p− ε2 (m+ )

g0(y) = Aη g1(ξ, y, ε) =

ε sin(q + t)ε cos(q + t)−ε cos(q + t)

.

Aiming at a discussion near the separatrices of the unperturbed system we focus on the region describedvia |p| ≤ and |η| ≤ for any large > 0 fixed. The vectorfield may be changed outside this domain(i.e. for large |p|, |η|) without any influence on the region investigated. More precisely it is possible tomodify the vector field in a way such that the ”new” maps f0, g0, f1 and g1 satisfy the assumptions madeon the boundedness and regularity in proposition 1.6.3. Note that the original vector field is periodicwith respect to t and q and hence of class BC r with respect to these variables.

In order to prove the existence of a global, attractive invariant manifold we show that we are in thesecond situation considered in the assumptions of proposition 1.6.3.

Using

Df0(ξ) =

0 0 00 0 1

0 −(

a2

)2cos(q) 0

and P =

1 0 00 1 10 a

2 −a2

the value

max|ξ|≤1ξ∈R

n

−(


= max|ξ|≤1q∈R

a4

(

q2 − p2)

(cos(q)− 1)

is found to be equal to a2 . Choosing the maximum norm on R3 the logarithmic norm of Dg0(y) is given

by µ (A) = maxi=1...n

ℜ(λ) |λ ∈ σ(A) = − 12 (cf. remark 1.6.2). Hence for any a < 1/r the existence of an

invariant manifold follows by consequence of proposition 1.6.3. The degree r of differentiability will beneeded to be at least equal to 2.


4.7.2 Partial Calculation of the Invariant Manifold

In this section we discuss the case where (4.14) admits a global, attractive invariant manifold. In muchthe same way as in section 1.6.4 we consider the reduced system, i.e. (4.14) restricted to the manifold.This reduced system is of perturbed pendulum type: if we denote the parametrization of the manifoldby S again then the reduced system reads

(q, p) = J∇H(q, p) + F (q, p,S(q, p, t, ε), t, ε). (4.59)

Using the differentiability of the map S with respect to ε we consider the representation

S(q, p, t, ε) = εS1(q, p, t) + ε2 S2(q, p, t, ε) (4.60)

where S1 is BC 1. Thus the equation of invariance (derived in a similar way as (1.147)) of S implies

∂tS1(q, p, t) + ∂(q,p)S1(q, p, t)J ∇H(q, p) = AS1(q, p, t) +G1(q, p, t). (4.61)

Aiming at the discussion of the region close to the separatrices of the unperturbed system of (4.59) weintroduce the Melnikov function M : denoting the (upper or lower) separatrix solution by (qs, ps) thecorresponding Melnikov function is given by

M(t, ε) =

∫

R

J ∇H(qs(s), ps(s)) ∧ F (qs(s), ps(s),S(qs(s), ps(s), s+ t, ε), s+ t, ε) ds (4.62)

(cf. e.g. formula 4.5.16 in [5]). We emphasize that in order to derive explicit formulae for this Melnikovfunction it suffices to calculate S(qs(s), ps(s), s+ t, ε) i.e. not the values of S for any (q, p) but evaluatedon the separatrix solutions (qs, ps) solely.

Hence we focus on the calculation of the quantity σ(τ, t, ε) := S(qs(τ), ps(τ), t, ε), in particular

σ1(τ, t) := S1(qs(τ), ps(τ), t).

In a first step we note that the identity ∂τ (qs(τ), ps(τ)) = J ∇H(qs(τ), ps(τ)) together with (4.61) implies

∂tσ1(τ, t) + ∂τσ

1(τ, t) = ∂tS1(qs(τ), ps(τ), t) + ∂(q,p)S1(qs(τ), ps(τ), t)J ∇H(qs(τ), ps(τ))

= Aσ1(τ, t) +G1(qs(τ), ps(τ), t). (4.63)

Since G1(q, p, t) =∑

|n|≤NG1n(q, p) e

int we may solve (4.63) by plugging the ansatz

σ1(τ, t) =∑

|n|≤Nσ1n(τ) e

int

into (4.63):

∑

|n|≤Ni n σ1

n(τ) eint +

∑

|n|≤N∂τσ

1n(τ) e

int = A∑

|n|≤Nσ1n(τ) e

int +∑

|n|≤NG1n(qs(τ), ps(τ)) e

int.

Comparing the Fourier coefficients then yields

∂τσ1n(τ) = [A− i n IR3 ]σ1

n(τ) +G1n(qs(τ), ps(τ)) (4.64)


and thus by the variation of constants formula

σ1n(τ) = eτ (A−i n I

R3)

σ1n(0) +

τ∫

0

e−s (A−i n IR3)G1

n(qs(s), ps(s)) ds

.

However since the map S is bounded, the same must be true for σ and hence for every coefficient mapσ1n, |n| ≤ N as well. Taking into account that all eigenvalues of the matrix A have negative real value

this implies

σ1n(0) =

0∫

−∞

e−s (A−i n IR3)G1

n(qs(s), ps(s)) ds

such that

σ1n(τ) =

τ∫

−∞

e(τ−s) (A−i n IR3 )G1

n(qs(s), ps(s)) ds. (4.65)

In view of the representations (4.21) and (4.22) found for the application of the miniature motor wetherefore conclude

σ11(τ) =

τ∫

−∞

eiqs(s) e(τ−s) (A−i IR3) v ds =

0∫

−∞

2a e

iqs(τ+2a t2) e−

2a t2 (A−i I

R3 ) v dt2

σ1−1(τ) =

τ∫

−∞

e−iqs(s) e(τ−s) (A+i IR3) v ds =

0∫

−∞

2a e

−iqs(τ+ 2a t2) e−

2a t2 (A+i I

R3 ) v dt2

(4.66)

and σ1n = 0 for |n| 6= 1. Setting

ι1(τ, t2) =2a e

iqs(τ+2a t2) e−

2a t2 (A−i I

R3) v and ι−1(τ, t2) =2a e

−iqs(τ+ 2a t2) e−

2a t2 (A+i I

R3) v

we will prefer the representation

σ11(τ) =

0∫

−∞

ι1(τ, t2) dt2 σ1−1(τ) =

0∫

−∞

ι−1(τ, t2) dt2 (4.67)

in what follows.


4.7.3 The Melnikov Function

In this subsection we derive a more explicit formula for the Melnikov function (4.62) for the applicationconsidered here. Using the identities (4.1.3), (4.20)–(4.22) we have

M(t, ε) =

∫

R

(

ps(s)

−(

a2

)2sin(qs(s))

)

∧

(

ε(

eiqs(s)M σ(s, s+ t, ε) ei(s+t) + e−iqs(s)M σ(s, s+ t, ε) e−i(s+t))

+ ε2(

0− ps(s)− (m+ )

)

)

ds.

Taking into account that due to the zeroes in the first row of the matrix M

M σ(s, s+ t, ε) = ε

(

0(

M2,.|σ1(s, s+ t))

)

+O(ε2)

(where M2,. denotes the second row of M) we have

M(t, ε) = ε2∫

R

ps(s)(

eiqs(s)(

M2,.|σ1(s, s+ t))

ei(s+t) + e−iqs(s)(

M2,.

∣

∣σ1(s, s+ t))

e−i(s+t)

− ps(s)− (m+ ))

ds+O(ε3).

As derived in the previous subsection 4.7.2 the representation

σ1(s, s+ t) = σ11(s) e

i(s+t) + σ1−1(s) e

−i(s+t)

applies. This implies

M(t, ε) = ε2∫

R

ps(s)(

eiqs(s)(

M2,.|σ1−1(s)

)

+ e−iqs(s)(

M2,.

∣

∣ σ11(s)

)

− ps(s)− (m+ ))

ds

+ε2∫

R

ps(s) eiqs(s)

(

M2,.|σ11(s)

)

ei2s ds ei2t

+ε2∫

R

ps(s) e−iqs(s) (M2,.

∣

∣σ1−1(s)

)

e−i2s ds e−i2t +O(ε3)

and hence

M(t, ε) = ε2(

m0 +m2 ei2t +m2 e

−i2t)+O(ε3) (4.68)

where

m0 =

∫

R

2a ps(

2a t1)

(

eiqs(2a t1)

(

M2,.|σ1−1(

2a t1)

)

+ e−iqs(2a t1)

(

M2,.

∣

∣σ11(

2a t1)

)

− ps( 2a t1)− (m+ )

)

dt1

m2 =

∫

R

2a ps(

2a t1) e

iqs(2a t1) ei

4a t1

(

M2,.|σ11(

2a t1)

)

dt1.


In view of the numerical computations to follow we enhance the formulae by using (4.67).

m0 = 4a

∫

R

0∫

−∞

ℜ(

ps( 2a t1) e

−iqs( 2a t1)

(

M2,.

∣

∣ ι1(2a t1, t2)

)

)

dt2 dt1

− 2a

∫

R

ps( 2a t1)

2dt1 − (m+ ) 2a

∫

R

ps( 2a t1)dt1

m2 = 2a

∫

R

0∫

−∞

ps( 2a t1) e

iqs(2a t1) ei

4a t1

(

M2,.| ι1( 2a t1, t2))

dt2 dt1.

(4.69)

We will use this representation of the coefficients for the computations described in section 4.7.5.

4.7.4 The Separatrix Solutions

In order to complete the preparations necessary to compute the Melnikov function we give the explicitrepresentations of the terms ps( 2

a t1) and eiqs(

2a t1) arising in the formulae (4.69) for m0 and m2. From the

well known formula sin(12qs(s)) = ± tanh(a2 s) where qs(s) ∈ (−π, π) for all s ∈ R we deduce cos(12qs(s)) =√

1− sin2(12qs(s)) = sech(a2 s) such that

eiqs(2a t1) =

(

cos(12qs(2a t1)) + i sin(12qs(

2a t1))

)2= sech2(t1) (1± i sinh(t1))

2.

Since the map H is a first integral for the separatrix solutions (qs, ps) and (qs(0), ps(0)) = (0,±a) we find

a2

2= H(0,±a) = H(qs(s), ps(s)) =

ps(s)2

2+(a

2

)2

(1− cos(qs(s))) =ps(s)

2

2+a2

2sin2(12qs(s))

eventually implying

ps( 2a t1) = ± a

√

1− sin2(12qs(2a t1)) = ± a sech(t1).


4.7.5 Numerical Results

In this last subsection we present the results found for the Melnikov function as in (4.68). The valuesm0 and m2 were calculated via numerical integration of the formulae (4.69). We have carried out thesecomputations for a = 0.1 and a = 0.54. As the parameters and m do not appear inside the integrals,these values were kept indeterminate for computations.

a = 0.1

upper separatrix lower separatrix

mupp0 ≈ −3.0− 6.6 − 6.2m mlow

0 ≈ 2.2 + 5.8 + 6.2m

mupp2 ≈ 0.9 · 10−10 − 0.79 · 10−10 i mlow

2 ≈ 0.5 · 10−10 + 0.4 · 10−10 i

-3

-2

-1

0

1

2

1 2 3 4 5 6t

Melnikov functions for a=0.1, rho=0, m=0

Figure 4.37: Graph of the leading ε2–term of the Melnikov functions for a = 0.1, = 0 and m = 0.The function for the upper separatrix is plotted black whereas the lower one is plotted in grey.

For this choice of a the upper Melnikov function is strictly negative while the lower Melnikov function isstrictly positive. In view of the definition (4.62) we see that this result implies that the manifolds U2,+

and U2,− are situated ”outside” the manifolds U1,− and U1,+ as sketched in figure 4.38.

Consider two points Aupp, Alow on the separatrices of the unperturbed pendulum which are symmetricwith respect to the q–axis and near the hyperbolic fixed point on the right (cf. figure 4.38). Then letB denote a parallelogram defined via the orthogonal lines through Aupp and Alow . Taking into accountthat the real parts of the eigenvalues of the unperturbed linearized system at the fixed point are of thesame absolute value, similar arguments as used in the third step of the proof of proposition 2.3.11 implythat the leading ε–terms of d

(

U1,+,U2,+)

and d(

U1,+,U2,−) are identical, provided that Aupp and Alow

have been chosen sufficiently close to the hyperbolic fixed point. From formula (4.5.11) in [5] we thendeduce that for any fixed t ∈ R the ratio of the distances d

(

U1,+,U2,+)

and d(

U1,−,U2,+)

at time t maybe approximated as follows:


2,+

U1,+

U2,-

U

U U2,+d( , )

1,-

1,-

U1,+d( , )

U1,+d( , )U

2,-

UU

A

2,+

Alow

upp

B q

Figure 4.38: Sketch obtained by considering the values of the Melnikov function

d(

U1,+,U2,+)

d (U1,−,U2,+)≈ d

(

U1,+,U2,−)

d (U1,−,U2,+)≈∣

∣Mlow(t, ε)∣

∣

|∇H(Alow)||∇H(Aupp)||Mupp(t, ε)| =

∣

∣Mlow(t, ε)∣

∣

|Mupp(t, ε)| ≈∣

∣mlow0

∣

∣

|mupp0 |

where ≈ denotes the equality of the leading ε–terms. Therefore the amount of solutions passing theq–axis may be evaluated qualitatively by considering the value

∣

∣mlow0

∣

∣

|mupp0 | ≈ 0.7 (for = m = 0)

and its dependence on the parameters a, and m. Numerical simulations confirm the situation foundanalytically, as depicted in figure 4.39.

2,+

U2,-

U

1,- U1,-

U1,+

U

q

Figure 4.39: Position of the stable and unstable manifolds for a = 0.1. This figure is taken from thephase portrait found by numerical integration using dstool (ε = 0.01).


a = 0.54

Although the parameter a must satisfy a < 0.5 (cf. section 4.7.1) to establish the existence of the global,attractive invariant manifold the integrals (4.69) converge for a = 0.54 as well and yield the followingresults.

upper separatrix lower separatrix

mupp0 ≈ −5.2− 8.4 − 6.2m mlow

0 ≈ 1.7 + 4.1 + 6.2m

mupp2 ≈ −1.6 · 10−9 + 1.6 · 10−9 i mlow

2 ≈ 6.1 · 10−3 − 11.0 · 10−3 i

-5

-4

-3

-2

-1

0

1

1 2 3 4 5 6t

Melnikov functions for a=0.54, rho=0, m=0

Figure 4.40: Graph of the leading ε2–term of the Melnikov functions for a = 0.54, = 0 and m = 0.The function for the upper separatrix is plotted black whereas the lower one is plotted in grey.

The situation for a = 0.54 is qualitatively equivalent to the case a = 0.1 explained above. This explainsthe behaviour found in section 4.2.2 for a = 0.54 where we have seen that some solutions pass the q–axisand tend towards p ≈ −1 while the remaining solutions are caught near p = 0. Evaluation of

∣

∣mlow0

∣

∣

|mupp0 | ≈ 0.3 (for = m = 0)

indicates that the ratio of solutions passing the q–axis is smaller than for a = 0.1 (cf. figure 4.41).


2,+

U2,-

U1,+

U

U

1,- q

Figure 4.41: Position of the stable and unstable manifolds for a = 0.54. This figure is taken from thephase portrait found by numerical integration using dstool (ε = 0.05).


4.8 Conclusion

In lemma 4.1.2 we considered the initial values of (4.1) which in a physical interpretation correspondto the switching on of the miniature synchronous motor. Although the corresponding initial value η2(0)is of size O(1/ε), all these solutions approach attractive invariant manifolds exponentially fast providedthat they start inside the upper, central or lower domain. The behaviour near these invariant manifoldsthen depends strongly on the parameter a.

For small a the set of these ”physical” initial values is entirely contained in the lower domain and thecorresponding solutions are attracted by the region p ≈ −1. This may be interpreted in a physical way asfollows: When switching on the electrical motor, after some transient state, the electrical circuit exhibitsa periodic behaviour. As for the mechanical system p ≈ −1 corresponds to d

dtϑ ≈ 0, the rotor oscillatesbut does not perform any rotation. If a sufficiently large linear damping is added (e.g. = 1) the rotorstill oscillates. In the case where a load is added (e.g. for m = 1) the rotor eventually exhibits a backwardrotation, i.e. the angular speed d

dtϑ is strictly negative.Considering the entire system (i.e. admitting any initial values) we observe additional interesting effects.These include capture of solutions in resonances in the central domain, which is equivalent to a rotationwith constant frequency but a significant variation of the angular speed. A further effect arises whenconsidering solutions with an initial angular speed d

dtϑ > 1 (i.e. starting the motor ”too fast”). Most

of the corresponding solutions then slow down to a regular rotation with angular speed ddtϑ ≈ 1. For a

small choice of the parameter a, however, the angular speed may decrease even more until the frequencyeventually becomes zero.Therefore, from a physical point of view, the choice of a small a is not satisfactory as the rotor does notenter the synchronous rotation desired.

If on the other hand a is choosen large, then most of the ”physical” initial values are contained in thecentral domain. The results found imply that even if the motor is started with a linear damping or a load,the corresponding solutions tend towards the unique, exponentially asymptotic stable periodic solution of(4.14) near the origin (q, p) = (0, 0). From a physical point of view, the choice of parameters correspondingto large a therefore is satisfactory as the rotor enters the synchronous rotation when switched on.

It is easy to verify that the ε–expansion of the periodic solution is given by

q(t, ε) = −ε2(

2sin(2 t) a2 + a2 − 16

(a2 − 16) a2+ 4

m+

a2

)

+O(ε3)

p(t, ε) = −ε2(

4cos(2 t)

a2 − 16

)

+O(ε3).

(4.70)

Since the rotor performs a rotation with an angular speed ddtϑ = 1 + p(t, ε), the resulting frequency

corresponds to the one given by the power supply where, by consequence of the terms sin(2 t) and cos(2 t)in (4.70), the angular speed d

dtϑ varies periodically with the second harmonic of the basic frequency anda small amplitude. This small, periodic variation of the angular speed is a well–known phenomena inelectrical engineering. We eventually not that by (4.70) again the size of the load and damping (given bythe parameters and m) determine the phase difference between the rotor and the magnetic field but donot influence the angular speed in the main.

Chapter 5

Application to Van der Pol’sEquation

5.1 Introduction

In this chapter we discuss the application of the results found in chapters 1 to 3 on the Van der Pol’s–likeequation1

x′′(τ) − α(

γ + x(τ)2)

x′(τ) + x(τ) = β cos(τ/a). (5.1)

We will discuss the situation where ε :=√aα may be chosen sufficiently small (i.e. for |ε| < ε1) and

there exists a parameter β such that β = ε2 β/a2 for all |ε| < ε1.

5.2 Transformations following Chapter 1

Setting

t := τ/a q(t) := x(t a) p(t) := x′(t a).

we rewrite (5.1) as follows:

q = a p

p = −a q + ε2[ (

γ + q2)

p+ β cos(t)]

.(5.2)

This system is of the form (1.1) although the η–subsystem does not appear and therefore may be set toη = −η. By consequence the discussion carried out in the previous chapters is much simpler and doesnot require any computational assistance as for the application presented in chapter 4.

The following steps may be established at once:

1for γ = 1 (5.1) corresponds to the Van der Pol system as considered in [5], Eq.2.1.1

229

230 Chapter 5. Application to Van der Pol’s Equation

1. General Assumption GA1 : The Hamiltonian system (1.2) corresponds to the harmonic oscil-lator and therefore satisfies GA 1.1 with J = R provided that Ω(p0) = a 6∈ Z. As the η–system isomitted, GA 1.2 and GA 1.4 do not apply here. The assumptions made in 1.97 a–1.97d are fulfilledby the map P(h) = h and finally the representations (1.3), (1.4) hold with

F 20 (q, p) =

(

0(

γ + q2)

p

)

F 2±1(q, p) =

(

0β/2

)

.

2. Periodic Solution : Applying the explicit formulae given in proposition 1.2.4 we compute

F 1 = 0 F 2(Q, P , t) =

(

0(

γ + Q2)

P

)

F 3(Q, P , t) = 0.

3. Strongly Stable Manifold : Due to the absence of the η–system in (5.2) the considerationsmade in section 1.4 are not necessary here. Therefore the change of coordinates (1.86) simplifies to(Q, P ) = (Q,P ) and hence

F 1 = 0 F 2(Q,P, t) =

(

0(

γ +Q2)

P

)

F 3(Q,P, t) = 0.

4. Action Angle Coordinates : As the transformation Φ constructed via the solutions of the

harmonic oszillator is given by Φ(ϕ, h) = h

(

sin(ϕ)cos(ϕ)

)

the definitions made in (1.110) read

F2(t, ϕ, h, ε) = a+ a1a

h

((

cos(ϕ)− sin(ϕ)

)∣

∣

∣

∣

ε2(

0(

γ + h2 sin2(ϕ))

h cos(ϕ)

)

+O(ε4; t, ϕ, h)

)

F3(t, ϕ, h, ε) =1a

h

(

h

(

sin(ϕ)cos(ϕ)

)∣

∣

∣

∣

ε2(

0(

γ + h2 sin2(ϕ))

h cos(ϕ)

)

+O(ε4; t, ϕ, h)

)

hence (1.111) may be written in the non–autonomous form

ϕ = a− sin(ϕ)[

ε2(

γ + h2 sin2(ϕ))

cos(ϕ) +O(ε4; t, ϕ, h)]

h = h1

acos(ϕ)

[

ε2(

γ + h2 sin2(ϕ))

cos(ϕ) +O(ε4; t, ϕ, h)]

.(5.3)

Note that the range L of Φ is the entire (q, p)–plane and h = 0 is an invariant set corresponding tothe unique 2π–periodic solution near the origin.

5. Attractive Invariant Manifold and the Reduced System : As the system (5.3) is alreadytwo dimensional, the considerations made in section 1.6 may be dropped here as well. The reducedsystem (1.158) is equal to (5.3).

5.3. Discussion of the Global Behaviour following Chapter 2 231

5.3 Discussion of the Global Behaviour following Chapter 2

Since (5.3) is already of the form (1.159) it remains to calculate the mean value g20,0 in this case. It mayreadily be seen that averaging the ε2 term of the second equation in (5.3) with respect to ϕ yields

g20,0(h) = h

(

γ

2 a+ h2

1

8 a

)

.

As the frequency ω(h) = a 6∈ Z remains constant for all h ∈ R, the set of resonant frequencies R is empty.Hence the global behaviour is determined by the drift g20,0(h) uniquely. We therefore conclude:

5.4 a. If γ ≥ 0 then for any > 0 the values of h are positive for all 0 < h ≤ provided that ε issufficiently small. Thus on any fixed bounded domain, ε may be chosen small in a way such thatall solutions (up to the ”periodic solution” h = 0) leave this domain as t→ ∞.

5.4 b. If γ < 0 then the map g20,0(h) admits a zero at h∗ = 2√

|γ|. The values g20,0(h) are positive forh > h∗ and negative for h < h∗. Given any large constant > 0 and a small ǫ > 0, ε may be chosensufficiently small such that all solutions with initial values 0 < h < h∗− ǫ approach the origin whileorbits starting with h∗ + ǫ < h ≤ tend towards h = .

5.4 Discussion of the Stability of the Set h = 0 following Chap-ter 3

Comparing the system (5.3) with (1.160) yields the following identities:

g,1(t, ϕ, ε) =1

acos(ϕ)

(

ε2 γ cos(ϕ) +O(ε4; t, ϕ, h))

g,2(t, ϕ, ε) = O(ε4; t, ϕ, h)

g,3(t, ϕ, ε) =1

acos(ϕ)

(

ε2 sin2(ϕ) cos(ϕ) +O(ε4; t, ϕ, h))

.

We therefore conclude:

5.5 a. If γ 6= 0 then g,10,0(ε) = ε2 γ2 a + O(ε4). Choosing ε sufficiently small, the invariant set h = 0 is

therefore (linear) stable if γ < 0 and unstable if γ > 0.

5.5 b. If γ = 0 then g,30,0(ε) = ε2 18 a +O(ε4). This implies the (cubic) instability of the invariant set h = 0

(for ε small).

5.5 Conclusion

Summarizing the results found on the global behaviour and the stability of the periodic solution of (5.2)we see that varying the parameter γ, system (5.2) admits a subcritical Hopf bifurcation at γ = 0 (whenomitting O(ε4)–terms). For the critical value γ = 0 the periodic solution (q, p) = O(ε) near the origin isunstable.

232 Chapter 5. Application to Van der Pol’s Equation

γ

h

stable

unstable

Figure 5.1: subcritical Hopf bifurcation for the Van der Pol–like sytem (5.1)

Bibliography

[1] H. Amann, Ordinary Differential Equations, De Gruyter studies in mathematics 13 (1990)

[2] P.F. Byrd & M.D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, SecondEdition, Springer (1971)

[3] DSTOOL Dynamical System Analysis Tool, Version 1.1(available at ftp://cam.cornell.edu/pub/dsool/)

[4] N. Fenichel, Persistence and Smoothness of Invariant Manifolds for Flows, Ind. Univ. Math. J.,Vol. 21, No. 3 (1971), 193–225

[5] J. Guckenheimer & P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations ofVector Fields, Third Printing, Springer (1990)

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[7] E. Kamke, Differentialgleichungen Bd.1, Teubner (1977)

[8] Al Kelley, The Stable, Center–Stable, Center, Center–Unstable, Unstable Manifolds, Journ. ofDiff. Eq. 3 (1967), 546–570

[9] U. Kirchgraber, F. Lasagni, K. Nipp, D. Stoffer, On the application of invariant manifold theory, inparticular to numerical analysis, Intern. Ser. of Num. Math., Vol. 97 (1991), 189–197

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[12] Laborberichte zum polarisierten Kondensatormotor (AMY12), Landis & Gyr, 1982

[13] K. Nipp & D. Stoffer, ETH Zurich, Switzerland, An Invariant Manifold Result, unpublishedmanuscript

[14] D. Stoffer, On the approach of Holmes and Sanders to the Melnikov procedure in the method ofaveraging, Journal of Applied Mathematics and Physics (ZAMP), Vol. 34, November 1983

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[17] J.A. Sanders & F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Springer (1985),section 5.5

[18] T. Strom, On logarithmic norms, SIAM Num. Anal., Vol. 12, No. 5, October 1975

233

Lebenslauf

Ich wurde am 3. Juli 1967 in Baden geboren und wuchs in Windisch auf, wo ich wahrend funf Jahren diePrimarschule und vier Jahre lang die Bezirksschule besuchte. Die dreieinhalb Jahre an der KantonsschuleBaden schloss ich im Herbst 1986 mit der Matura Typ C (naturwissenschaftliche Richtung) ab. Nacheinem Zwischenjahr in der Industrie begann ich im Wintersemester 1987/88 mein Studium der Mathe-matik mit Nebenfachern Physik und Informatik an der Universitat Zurich, wo ich im November 1992diplomierte. Seit 1993 arbeite ich als Assistent an der Abteilung IX fur Mathematik und Physik an derETH Zurich.

Danksagung

Abschliessend mochte ich einigen Personen danken, welche mir auf meinem bisherigen akademischenWegebegegnet sind:

Meinem Gymnasiallehrer Dr. F. Naf verdanke ich meinen Entschluss zur Mathematik (in Harmonie mitder Musik). Durch seine begeisternden Lektionen vermochte er mir die Schonheit und Faszination derMathematik zu eroffnen.Prof. Dr. H. Amann zeigte mir wahrend meines Studiums die eleganten, kraftvollen und umfassendenFarben, in denen die Mathematik zu erscheinen vermag. Seine Klarheit und Ehrlichkeit bleibt mir einVorbild.Meinem Doktorvater Prof. Dr. U. Kirchgraber bin ich fur seine Einladung, an der ETH zu promovierenund mathematisch zu reifen, zu grossem Dank verpflichtet. Er liess mich die Geissel, die durch dieNotwendigkeit des Gebrauchs der strengen mathematischen Sprache gegeben ist, spuren und bot mirGelegenheit, Ruhe und Geduld zu uben.Von der Diplomarbeit bis zu dieser Dissertation hatte mein Zimmergenosse PD Dr. ”Danu” Stoffer stetsein offenes Ohr fur Ideen und Fragen. Die unzahligen Diskussionen und Anregungen, die ich ihm zuverdanken habe, fuhrten mich oft genug auf den richtigen Weg zuruck.Fur sein aufrichtiges Interesse mochte ich meinem Koreferenten Prof. Dr. E. Zehnders danken. Seineerfrischenden Kommentare motivierten mich nachhaltig.Des weiteren danke ich Dr. K. Nipp und Dr. D. Stoffer fur ihre anwenderfreundliche Formulierung desIM–Satzes, Prof. Dr. J. Waldvogel fur seine Anregungen zum vierten Kapitel und L. Bernardin fur diefreundliche Unterstutzung bei meiner Arbeit mit Maple.

Den zahlreichen Familienangehorigen, Freunden und Mitarbeitern, die mich neben dem akademischenLeben begleitet und unterstutzt haben, gebuhrt schliesslich ein ganz herzliches ”Dankeschon”! Euchallen mochte ich diese Arbeit widmen, als Zeichen dafur, dass ich Euren Beistand, ohne den ich diese Aranicht zu Ende gefuhrt hatte, stets geschatzt habe.

Experience is not what happens to you.It is what you do with what happens to you.

Aldous Huxley, 1894–1963

Documents

Invariant Manifolds, Passage through Resonance, Stability and a Computer Assisted Application to a Synchronous Motor